SPECTRAL AND SCATTERING THEORY FOR SYMBOLIC POTENTIALS OF ORDER ZERO

ANDREW HASSELL^{∗}, RICHARD MELROSE^{†}, AND ANDRÁS VASY^{‡}

Abstract. The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on ℝ^{2} which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at noncritical energies are shown to originate both at minima and maxima, although the latter are not germane to the L^{2} spectral theory. Asymptotic completeness is shown, both in the traditional L^{2} sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.

Introduction

In [7], Herbst initiated the spectral analysis and scattering theory for the Schrödinger operator P = Δ+V on ℝ^{n} where V is a smooth, real-valued potential which is homogeneous of degree zero near infinity. Further results for such potentials were obtained by Agmon, Cruz and Herbst [1] and Herbst and Skibsted [6]. Such a potential is so large that it deforms the geometry near spatial infinity and consequently the scattering theory of P is quite different from that of a Schrödinger operator with a short-, or even a long-, range potential. Herbst [7] and Herbst-Skibsted [6] showed that solutions of the time-dependent Schrödinger equation, with L^{2} initial data, concentrate for large times near critical directions of the potential restricted to the sphere at infinity.

In this paper we study in some detail the tempered eigenspaces of P, consisting for each fixed λ of those functions u, of polynomial growth, satisfying (P −λ)u = 0. We restrict attention to the two- dimensional case but generalize from Euclidean space, ℝ^{2}, to scattering metrics (defined below) on arbitrary compact 2-manifolds X with boundary. Such metrics give the interior of X the structure of a complete Riemannian manifold with curvature vanishing at infinity. In this wider context the natural condition replacing, and generalizing, homogeneity of the potential near infinity is its smoothness up to the boundary of ∂X. For the Euclidean case, now appearing through the radial compactification to a ball, this allows potential which are classical symbols of order 0. Generally we assume that V_{0}, the restriction of the potential to the boundary, is Morse, so V_{0} has only nondegenerate critical points.

1991 Mathematics Subject Classification. 35P25, 81Uxx.

Key words and phrases. scattering metrics, degree zero potentials, asymptotics of generalized eigenfunctions, Microlocal Morse decomposition, asymptotic completeness.

∗ Supported in part by the Australian Research Council, ^{†} Supported in part by the National Science Foundation under grant #DMS-9622870, ^{‡} Supported in part by the National Science Foundation under grant #DMS-99-70607.

The scattering wavefront set from [13] allows the eigenfunctions of P to be analyzed microlocally and hence to be related to the classical dynamics of P. It is the asymptotic behaviour which is relevant here and this is described through the contact geometry of the cotangent bundle over the boundary; the classical dynamics of P are then given by a Legendre vector field over the boundary. The critical points of this vector field correspond to the critical points of V_{0}. We give microlocal versions of the familiar notions of incoming and outgoing eigenfunctions, depending on which integral curves of the Legendre vector fields are permitted in the scattering wavefront set. By direct construction we show that there is a non-trivial space of outgoing microlocal eigenfunctions associated to each critical point q of V_{0}, at energies λ above V_{0}(q). For a minimum this space is isomorphic to the space of Schwartz functions on a line, for a maximum it is isomorphic to the space of formal power series (i.e. Taylor series at a point) in one variable. These isomorphisms are realized directly in terms of asymptotic expansions near the corresponding critical point of V_{0}; the form of the expansion reflects the behaviour of the classical trajectories near these points.

For non-critical eigenvalues there is a well-defined space of ‘smooth’ eigenfunctions, which may be characterized either as the range of the spectral projection on Schwartz functions or else as the space of tempered eigenfunctions with no purely tangential oscillation at infinity (this latter condition is conveniently expressible in terms of the scattering wavefront set). We show this space to have a ‘microlocal Morse decomposition’ in terms of the spaces of microlocally outgoing eigenfuncitons associated to the critical points. The space of all tempered eigenfunctions is identified by a boundary (or Green) pairing with the dual of the space of smooth eigenfunctions, so it has a similar decomposition in terms of the duals of the microlocal spaces. The latter are isomorphic either to Schwartz distributions on a line (for minima) or polynomials in one variable (for maxima). The map from generalized eigenfunctions to the duals of the microlocally outgoing eigenspaces thus correspond to ‘generalized boundary data’; the collective boundary data fixes the eigenfunction via the inverse map, which is an analogue of the Poisson operator. There are two such families of boundary data (corresponding to an underlying choice of orientation of the Legendre vector field) and the isomorphism between them is the scattering matrix.

These results are new even in the Euclidean setting. Although for the most part the results here are limited to the two-dimensional case there are obvious generalizations to higher dimensions. These extensions will be discussed in future publications as will the related microlocal description of the Poisson operators and scattering matrix.

Superficially, the existence of smooth eigenfunctions associated to a maximum might appear to be in conflict with the results of Herbst and Skibsted on decay, near maxima, of solutions to the time-dependent Schrödinger equation [6]. This is not the case, since these eigenfunctions are, near the maximum point, in a weighted L^{2} space which implies that they are too small to violate this decay estimate. In fact an eigenfunction which is microlocally outgoing near a maximum, and is nontrivial there, is necessarily non-trivial (but not outgoing) near the minimum. Thus there are no eigenfunctions purely associated to a maximum, rather these outgoing eigenfunctions can be thought of as corresponding to the completion of the Schwartz functions on a line to smooth functions on a circle by the addition of the Taylor series at the point at infinity. When completed in a norm equivalent to the L^{2} norm on initial data for the Schrödinger operator, these extra terms are already in the closure of outgoing eigenfunctions at the minimum, thus they do not ‘appear’ in the L^{2} theory.

Next we give a more precise description of our results. Thus, let X be a compact manifold with boundary where, for the moment, we do not restrict the dimension. The boundary Y = ∂X consists of a finite union of compact manifolds without boundary, Y = Y_{1} ∪ ⋯ ∪ Y_{N}. It is always possible to find a boundary defining function on X, x ∈ C^{∞}(X) such that x ≥ 0, Y = {x = 0} and dx ≠ 0 on Y. A Riemannian metric on the interior of X is a scattering metric if, for some choice of defining function, it takes the form

(I.1)  g = \frac{dx^{2}}{x^{4}} + \frac{h}{x^{2}} , h ∈ C^{∞}(X; S^{2}X), h_{0} = h|_{Y} positive definite.

That is, h = x^{2}(g − x^{−4}dx^{2}) is a smooth 2-cotensor on X which restricts to a metric on Y. In this setting we consider a real potential V ∈ C^{∞}(X) and examine the spectral and scattering theory of Δ + V where Δ is the Laplace operator of a scattering metric. The Euclidean case is included since the Euclidean metric is a scattering metric on the radial compactification of ℝ^{n} to a ball (or half-sphere) and a smooth potential which is homogeneous of degree zero near infinity is a smooth function up to the boundary of the radial compactification. The lower order parts of the Taylor series of V allow additional terms decaying at Coulomb rate or faster.

Under these assumptions, for dim X ≥ 2, the Schrödinger operator P = Δ + V is self-adjoint and has continuous spectrum of infinite multiplicity occupying the interval [κ, ∞) where

(I.2)  κ = inf_{Y} V.

In addition there may be point spectrum in the interval [m, K],

(I.3)  m = inf_{X} V, K = sup_{Y} V

which is discrete in the open set (−∞, K) \ Cv(V),

(I.4)  Crit(V) = {p ∈ Y; d_{Y} V (p) = 0}, Cv(V) = V (Crit(V)).

The main interest lies in the continuous spectrum which we analyse in detail here under the assumption that dim X = 2 and V_{0} = V |_{Y} is Morse. For simplicity in this Introduction we outline the results under the additional assumption that Y has only one component circle and V_{0} is perfect Morse, so only has a global maximum, z_{max}, and minimum z_{min}, forming Crit(V). These restrictions are removed in the body of the paper.

Our central result is the parameterization of all tempered distributions associated to the continuous spectrum; this constitutes a distributional form of ‘asymptotic completeness’. Thus we examine

(I.5)  E(λ) = {u ∈ C^{−∞}(X); (Δ + V − λ)u = 0} .

The space of extendible distributions, C^{−∞}(X), reduces precisely to the space of tempered distributions in the sense of Schwartz in case X is the radial compactification of ℝ^{n}. For eigenfunctions this is equivalent to a polynomial bound. For λ < κ, E(λ) is finite-dimensional and consists of square-integrable eigenfunctions. In fact, for any λ ∉ Cv(V), we show in Proposition 4.9 that E_{pp}(λ) = E(λ) ∩ L^{2}(X) is contained in Ċ^{∞}(X) and finite dimensional (where L^{2}(X) is computed with respect to the Riemannian volume form) and is trivial for λ > K. Since C^{−∞}(X) is the dual of Ċ^{∞}(X) this allows us to consider

(I.6)

E_{ess}^{−∞}(λ) = {u ∈ C^{−∞}(X); (Δ + V − λ)u = 0, ⟨u, v⟩ = 0 ∀ v ∈ E_{pp}(λ)} ,

E(λ) = E_{ess}^{−∞}(λ) ⊕ E_{pp}(λ).

The structure of the space E_{ess}^{−∞}(λ) depends on λ; there are three distinct cases, corresponding to the values of κ, K and the additional transition point

(I.7)  λ_{Hess} = κ + 2V′′(z_{min}), κ = V (z_{min})

where the derivatives are with respect to boundary arclength. Clearly λ_{Hess} > κ but the three possibilities λ_{Hess} < K, λ_{Hess} = K, λ_{Hess} > K may all occur for different V (or for the same V but different metrics). For any particular problem only two or three of the following four intervals can occur

(I.8)  \begin{cases}κ < λ < min(λ_{Hess}, K) & “Near minimum” \\ λ_{Hess} < λ < K & “Hessian range” \\ K < λ < λ_{Hess} & “Mixed range” \\ max(λ_{Hess}, K) < λ & “Above thresholds.” \end{cases}

The critical values κ and K of V_{0} are thresholds corresponding to changes in the geometry of the fixed energy (i.e. characteristic) surface of the classical problem. Above these thresholds both the maximum and the minimum correspond to zeroes, which we also call radial points, of the classical flow (at infinity); the Hessian transition corresponds to an energy at which there is a change in the local geometry at the radial points corresponding to z_{min}.

To analyse E_{ess}^{−∞}(λ) we use the limit of the resolvent on the spectrum. This exists as an operator for λ ∈ (κ, +∞) \ (Cv(V) ∪ σ_{pp}(Δ + V)), as a consequence of an appropriate version of the Mourre estimate, or of microlocal estimates closely related to it, see [7] for the proof in the Euclidean setting.

Theorem I.1 (Herbst [7], see also Theorem 3.3). The resolvent

(I.9)  R(λ ± it) = (Δ + V − (λ ± it))^{−1}, t > 0, λ ∉ Cv(V)

extends continuously to the real axis, i.e. R(λ ± i0) exist, as bounded operators

(I.10)  x^{1/2+δ}L^{2}(X) ⊖ E_{pp}(λ) ⟶ x^{−1/2−δ}L^{2}(X) ⊖ E_{pp}(λ), ∀ δ > 0.

Note that

(I.11)  f ∈ x^{1/2+δ}L^{2}(X) ⊖ E_{pp}(λ) ⟹ [R(λ + i0) − R(λ − i0)]f ∈ E_{ess}^{−∞}(λ)

and conversely (essentially by Stone’s theorem) the range is dense.

For λ ∉ Cv(V) we can thus define spaces of ‘smooth’ eigenfunctions E_{ess}^{∞}(λ) by

(I.12)  E_{ess}^{∞}(λ) = [R(λ + i0) − R(λ − i0)](Ċ^{∞}(X) ⊖ E_{pp}(λ))

and E_{ess}^{∞}(λ) inherits a Fréchet topology from Ċ^{∞}(X). We give a microlocal characterization of these spaces below. In all of the non-transition regions

(I.13)  E_{ess}^{∞}(λ) ⊂ E_{ess}^{−∞}(λ) is dense

in the topology of C^{−∞}(X). In fact, approximating sequences can be constructed rather explicitly by extending R(λ±i0) to distributions satisfying a scattering wave front set condition; see Proposition 3.13.

One of the aims of scattering theory is to give explicit, and geometric, parameterizations of the continuous spectrum. The reality of the Laplacian implies that there are two equivalent such representations, which are interchanged by complex conjugation, and the scattering matrix gives the relation between them. The main task here is to describe the structure of the smooth eigenfunctions in the four nontransition regions in (I.8); our explicit parameterizations arise as the ‘leading terms’ in the complete asymptotic expansions that these eigenfunctions possess. This also identifies E_{ess}^{∞}(λ) in terms of familiar Fréchet spaces. The expansions are intimately connected to the corresponding classical problem, which is described in detail in §1. For λ ∉ Cv(V) the classical system consists of a Legendre vector field W (so defined up to a conformal factor) on a compact hypersurface, the characteristic variety Σ(λ), in a contact manifold. The dual variable to the variable 1/x (which is the radial variable r = |z| in the Euclidean case) is a function, ν on Σ(λ) with W ν ≥ 0.

The microlocal structure of this problem, near each radial point, when transformed from the scattering to the traditional context (essentially by Fourier transformation) is the problem considered by Guillemin and Schaeffer in [5]. Although we do not use their work explicitly here, several of our results could be proved by their techniques, provided one made the additional assumption of non-resonance of the linearization of Δ + V − λ at the radial points.

The projections of the radial points (that is, the zeroes of W) are the critical points of V on the boundary. Each such critical point z ∈ Crit(V) corresponds to two radial points q_{±} ∈ Σ(λ) with ±ν(q_{±}) > 0 if λ > V (z); it corresponds to a singular point on Σ(λ) if λ = V (z). The two points correspond to opposite, nonzero, values of ν. We shall denote by RP_{+}(λ) ⊂ Σ(λ), resp. RP_{−}(λ) ⊂ Σ(λ), the set of radial points at which ν > 0, resp. ν < 0. This can also be identified with the subset {q ∈ Crit(V); V (q) < λ} and we let

(I.14)  RP_{+}(λ) = Max_{+}(λ) ∪ Min_{+}(λ)

be the decomposition into points associated to maxima and to minima of V_{0}. We shall describe points in RP_{+}(λ), resp. RP_{−}(λ), as outgoing, resp. incoming, radial points.

Our examination of the structure of the eigenfunctions is based on the description of the microlocally outgoing eigenfunctions associated to each of the outgoing radial points. The notion of microlocality here is with respect to the scattering wavefront set, which is the notion of wavefront set associated to the scattering calculus in [13]. By microlocal elliptic regularity, for any u ∈ C^{−∞}(X), and any open set O

WF_{sc} ((Δ + V − λ)u) ∩ O = ∅ ⟹ WF_{sc}(u) ∩ O ⊂ Σ(λ).

If q ∈ RP_{±}(λ) and O is a sufficiently small open neighbourhood of q, which meets Σ(λ) in a W -convex neighbourhood with each W curve meeting ν = ν(q) we set

(I.15) Ẽ_{mic,+}(O, λ) = {u ∈ C^{−∞}(X); O ∩ WF_{sc} ((P − λ)u) = ∅

and WF_{sc}(u) ∩ O ⊂ {ν ≥ ν(q)}},

with Ẽ_{mic,−}(O, λ) defined by reversing the inequality. We may consider this as a space of microfunctions, E_{mic,+}(q, λ), by identifying elements when they differ by functions with wavefront set not meeting O. The result is then independent of the choice of O. That is, such microlocal solutions are determined by their behaviour in an arbitrarily small neighbourhood of q. Our convention is to call elements of either Ẽ_{mic,+}(q, λ) or E_{mic,+}(q, λ) ‘microlocally outgoing eigenfuncions at q ∈ RP_{+}(λ).’ The structure of the microlocal eigenfunctions at a radial point is essentially determined by the linear part of W, the normalized eigenvalues of which are given by (1.25) below. The character of the radial point is determined by the quantity

(I.16)  r(λ) = r(q, λ) = \frac{1}{2} − \sqrt{\frac{1}{4} − \frac{V′′_{0}}{2ν̃^{2}}} , ν̃ = ν(q),

where V′′_{0} is the Hessian with respect to boundary arclength at the critical point.

The case q ∈ Max_{+}(λ) is the most straightforward, with W having a saddle point at q at which r(q, λ) < 0. Elements u ∈ E_{mic,+}(q, λ) are then (i.e. have representatives which are) functions of the form

(I.17)  u = e^{iϕ_{λ}/x}x^{\frac{1}{2} +ϵ}v, v ∈ L^{∞}(X) ∩ C^{∞}(X^{∘}), ϵ = −\frac{1}{2}r(q, λ) > 0,

i.e. Legendre distributions (see [15]) where ϕ_{λ} ∈ C^{∞}(X) is a real function parameterizing the unstable manifold for W at q and v has a complete asymptotic expansion with C^{∞} coefficients as x → 0 in powers of x starting with a term in which the power is imaginary. Appropriately evaluating the terms in the expansion of v at z = π(q) gives a parameterizing map,

(I.18)  M_{+}(q, λ) : E_{mic,+}(q, λ) ∋ u ⟶ ℂ[[x]],

which is an isomorphism to the space of formal power series in one variable, or equivalently the space of arbitrary sequences s : ℕ ⟶ ℂ. This result is proved in Theorem 9.9. Notice that the factor x^{ϵ} in (I.17) means that these microlocal eigenfunctions are ‘small’ near the maximum, as they lie in the space x^{− \frac{1}{2}} L^{2}(X) locally near z. Here, the exponent −1/2 is critical for eigenfunctions; any eigenfunction which is globally in x^{− \frac{1}{2}} L^{2}(X) is actually rapidly decreasing.

When q ∈ Min_{+}(λ) the microlocal space E_{mic,+}(q, λ) can be identified with a global space of eigenfunctions

(I.19)  E_{mic,+}(q, λ) ≃ {u ∈ E_{ess}^{−∞}(λ); WF_{sc}(u) ∩ {ν > 0} ⊂ {q}} ⊂ E_{ess}^{∞}(λ)

(see Corollary 5.7). In this case, the structure of E_{mic,+}(q, λ) undergoes a distinct change as λ crosses the Hessian threhold. First consider the ‘near minimal’ range κ < λ < λ_{Hess}. In this case W has a center at q with Re r(q, λ) = \frac{1}{2}, and the square root in (I.16) is imaginary. It is convenient to introduce the space [X; z]_{\frac{1}{2}} obtained by parabolic blow up, with respect to the boundary, of X at z = π(q) (see section 2 for a discussion of blowups). Then the space C_{ff}^{∞}([X; z]_{\frac{1}{2}} ) consists of the smooth functions on this blown-up space which vanish rapidly at all boundary faces other than the ‘front face’ created by the blowup. Each microlocal eigenfunction has a representation in the form of an infinite sum

(I.20)  \begin{matrix}u = e^{iν̃/x} ∑_{j≥0} x^{\frac{1}{4} +i((2j+1)α+V_{1}(z))}a_{j}v_{j}, a_{j} ∈ S(ℕ), v_{j} ∈ C_{ff}^{∞}([X; z]_{\frac{1}{2}} ), \\ ν̃ = \sqrt{λ − V_{0}(z)}, V_{1}(z) = (∂_{x}V)(z), \end{matrix}

where the a_{j}, as indicated, form a Schwartz series in j, and the v_{j} restricted to ff([X; z]_{\frac{1}{2}} ) are L^{2}-normalized eigenfunctions for a self-adjoint globally elliptic operator on ff([X; z]_{\frac{1}{2}} ), namely a harmonic oscillator with eigenvalues (j + 1/2)α. The sequence of coefficients, a_{j}, is an arbitrary Schwartz sequence and the map obtained by normalization from (I.20) gives an isomormphism (I.21)

M_{+}(q, λ) : E_{mic,+}(q, λ) ⟶ Ċ^{∞}(ff[X; z]_{\frac{1}{2}} ), u ⟼ ∑_{j≥0}(a_{j}v_{j}|_{ff}), κ < λ < λ_{Hess},

onto the Schwartz functions on a line. We prove this statement in Theorem 7.4. Note that the normalization of v_{j} is only well-defined up to a factor e^{is(j)}, where s(j) is a real sequence. When it is analysed in a subsequent paper, it will convenient to introduce such terms so that M_{+}(q, λ) (or more precisely its inverse) will be a Fourier integral operator of an appropriate type.

The case λ_{Hess} ≤ λ, at a minimum, is similar, except that the discrete spectrum of the model problem has collapsed, and q has become a sink for W. At λ = λ_{Hess} the ‘local homogeneity’ is still parabolic as it is for λ < λ_{Hess}. As λ increases the local homogeneity is determined by r = r(q, λ) given by (I.16). Corresponding to this homogeneity we introduce the space [X; z]_{r} obtained by blow up with the correct scaling of the tangential variable relative to the normal variable. Now r is not necessarily rational, hence we need to replace C_{ff}^{∞}([X; z]_{r}) by a space of polyhomogeneous functions allowing both homogeneities x^{n} and x^{nr} where n is a non-negative integer. This space, A_{phg,ff}^{I}([X; z]_{r}), is described in more detail in Section 8, but here we mention that its elements are in particular continuous on [X; z]_{r}, and vanish at the boundary away from ff . If 1/r(λ) is not an integer, then the microlocal eigenfunctions u ∈ E_{mic,+}(q, λ) take the form

(I.22)  u = e^{iϕ_{λ}/x}x^{\frac{1−r}{2} +ic}a, a ∈ A_{phg,ff}^{I}([X; z]_{r}), r = r(λ),

where ϕ_{λ} ∈ C^{∞}(X) is now a real function parameterizing a smooth (minimal) Legendre submanifold through q and c = c(λ) is a real constant defined from the subprincipal symbol of P. Again, restriction to the front face of the blown-up space gives an isomorphism

(I.23)  M_{+}(q, λ) : E_{mic,+}(q, λ) ⟶ Ċ^{∞}(ff[X; z]_{r}), q ∈ Min_{+}(λ),

which, after selection of coordinates, maps isomorphically onto the Schwartz functions on an associated line. The passage of λ across the Hessian threshold results in a change in the underlying parameterizing space, which from then onwards depends on λ. This result is proved in Theorem 8.6, together with a statement of the minor changes needed when 1/r(λ) is an integer.

The space of smooth eigenfunctions can be directly related to these microlocal spaces and this is especially simple in case the potential is perfect Morse on the one boundary component. There is then a restriction map ‘to the maximum’

(I.24)  E_{ess}^{∞}(λ) ⟶ E_{mic,+}(q, λ), q ∈ Max_{+}(λ) for K < λ.

Theorem I.2. [Microlocal Morse decomposition] If dim X = 2, ∂X is connected, V_{0} is perfect Morse and λ ∉ Cv(V), the map (I.19) is an isomorphism

(I.25)  E_{mic,+}(q_{min}^{+}, λ) ⟶ E_{ess}^{∞}(λ), for κ < λ < K,

whereas, for λ > K, it combines with the map (I.24) to give a short exact sequence

(I.26)  0 ⟶ E_{mic,+}(q_{min}^{+}, λ) ⟶ E_{ess}^{∞}(λ) ⟶ E_{mic,+}(q_{max}^{+}, λ) ⟶ 0.

Whilst (I.26) does not split, there is a left inverse on any finite dimensional subspace. Combined with the isomorphisms (I.18) and (I.23) this result gives a quite complete description of the smooth eigenfunctions, which we could call ‘smooth asymptotic completeness at energy λ’ in view of (I.12).

There is a non-degenerate sesquilinear ‘Green pairing’ on E_{ess}^{∞}(λ) which extends to a continuous bilinear map (see section 12)

(I.27)  B : E_{ess}^{∞}(λ) × E_{ess}^{−∞}(λ) ⟶ ℂ

allowing E_{ess}^{−∞}(λ) to be identified with the dual space of E_{ess}^{∞}(λ). This pairing also restricts to a pairing between E_{mic,+}(q, λ) and E_{mic,−}(q, λ) for each radial point q and λ > V (π(q)). The adjoint of the map (I.18) becomes an isomorphism

(I.28)  ℂ[x] ⟷ E_{mic,−}(q, λ), q ∈ Max_{+}(λ),

where ℂ[x] is the space of finite power series (i.e. polynomials) — see Theorem 12.5. The range space in (I.28) again consists of functions which are Legendre distributions near q; only the trivial elements of E_{mic,−}(q, λ) are in the space x^{− \frac{1}{2}} L^{2}(X) near π(q) (this is dual to the property (I.17)). Similarly the adjoint of (I.23) extends to an isomorphism

(I.29)  S′(ℝ_{r}) ⟷ E_{mic,−}(q, λ), q ∈ Min_{+}(λ).

In the perfect Morse case (with ∂X connected) microlocal decompositions of the distributional eigenspaces are given by the identification

(I.30)  E_{mic,−}(q_{min}^{+}, λ) ⟶ E_{ess}^{−∞}(λ), for κ < λ < K

and for λ > K the short exact sequence

(I.31)  0 ⟶ E_{mic,−}(q_{max}^{+}, λ) ⟶ E_{ess}^{−∞}(λ) ⟶ E_{mic,−}(q_{min}^{+}, λ) ⟶ 0.

Combined with the isomorphism (I.28) and (I.29) this gives a decomposition of the full distributional eigenspaces in terms of microlocal eigenspaces.

One can compare the sequence in (I.26) with the familiar short exact sequence

(I.32)  S(ℝ) ↪ C^{∞}(𝕊) ⟶ ℂ[[x]]

where the second map is the passage to Taylor series at a point and the first is by identification of the complement of this point with a line. Then (I.31) corresponds to the dual sequence for distributions

(I.33)  ℂ[x] ⟶ C^{−∞}(𝕊) ⟶ S′(ℝ)

where the polynomials correspond to the (Dirac delta) distributions at the point.

In this way we may think of E_{ess}^{∞}(λ) as a space of test functions and E_{ess}^{−∞}(λ) as the dual space of distributions. It is then natural to think of the intermediate ‘Sobolev spaces’ of eigenfunctions. In this case they may be defined directly:

(I.34)  E_{ess}^{s}(λ) = {u ∈ E_{ess}^{−∞}(λ); WF_{sc}^{0,s−1/2}(u) ∩ {ν = 0} = ∅}, s ∈ ℝ.

The case s = 0 is particularly interesting since it is related to the notion of asymptotic completeness. In Section 12 we prove the following theorem.

Theorem I.3 (Asymptotic completeness at energy λ). If λ ∉ Cv(V) then

(I.35)

E_{ess,+}(q_{min}^{+}, λ) ↪ E_{ess}^{0}(λ) is dense and
M_{+}(q_{min}^{+}, λ) extends to a unitary operator E_{ess}^{0}(λ) ⟶ L^{2}(ℝ_{r}),

with respect to the metric induced by B from (I.27) on the one hand, and L^{2}(ℝ_{r}) equipped with a translation-invariant measure induced by h_{0} from (I.1) on the other.

Integration with respect to λ gives the usual version of asymptotic completeness which we state in Theorem 12.13.

These two theorems, combined with (I.17), have immediate implications for the ‘size’ of u_{±} = R(λ±i0)f, f ∈ Ċ^{∞}(X), away from the minima of V_{0}. Namely, for any pseudodifferential operator Q with q_{min}^{+} ∉ WF′_{sc}(Q), in particular for multiplication operators supported away from z_{min}, Qu_{+} ∈ x^{s}L^{2}(X) for all s < (−r(q_{min}^{+}, λ) − 1)/2. Since r(q_{min}^{+}, λ) < 0, this is an improvement over the statement that Qu_{+} ∈ x^{−1/2}L^{2}(X). The statement Qu_{+} ∈ x^{−1/2}L^{2}(X) can be interpreted as the absence of channels at the maxima of V_{0} and is closely related to the work of Herbst and Skibsted [6]; indeed this can be seen from the injectivity of M_{+}(λ) in (I.35). Thus, on the one hand, our results strengthen theirs in this special case (i.e. when dim X = 2) by showing that u_{+} has additional decay away from the minima of V_{0} and gives the precise asymptotic form of u_{+}. On the other hand, it also shows that the decay is not rapid. One interpretation of this phenomenon, supported by the short exact sequence (I.26) and isomorphism (I.35), is that the L^{2}-theory does not ‘see’ the maxima of the potential (because u_{+} is too small there), while the smooth theory (working modulo Ċ^{∞}(X)) does.

As well as the identification (I.35) arising from the outgoing parameterization there is the corresponding incoming parameterization, corresponding to ν < 0. The scattering matrix is then the unitary operator given by the composite

(I.36)  S(λ) = M_{+}M_{−}^{−1} : L^{2}(ℝ_{r}) → L^{2}(ℝ_{r}).

In a future publication it will be shown to be a Fourier integral operator, of an appropriate type, associated with the limiting scattering relation arising from the vector field W.

The classical dynamical system underlying the eigenvalue problem is analyzed in § 1, and § 2 contains a brief description of the blow-up procedure used later to discuss expansions. In § 3 extension of results of Herbst on the limiting absorption principle to the present more general setting are discussed; they are proved by positive commutator methods in § 4. The basic properties of the (incoming and outgoing) microlocal eigenfunctions associated to critical points of the classical system (i.e. radial points) are presented in § 5 and an outline of the methods used to analyse them is given in § 6. The next four sections contain the detailed analysis of incoming microlocal eigenfunctions at radial points associated to minima and both incoming and outgoing eigenfunctions associated to maxima. The decomposition of smooth eigenfunctions in terms of these microlocal components is described in § 11 and this is used, together with the natural pairing on eigenfunctions, to describe all tempered eigenfunction, the scattering matrix and various forms of asymptotic completeness in the final section.

The authors are delighted to acknowledge conversations with Ira Herbst and Erik Skibsted, who were working on closely related problems at the time, and Gunther Uhlmann for useful advice. They are also grateful for the hospitality and support of the Erwin Schrödinger Institute in Vienna, Austria, where some of the work on this paper was carried out.

1. Classical problem

The classical system formally associated to the Schrödinger operator Δ + V on ℝ^{n} is generated by the Hamiltonian function

(1.1)  p = |ζ|^{2} + V (z) ∈ C^{∞}(ℝ_{z}^{n} × ℝ_{ζ}^{n}).

In fact, only the ‘large momentum’ (the usual semiclassical limit) ζ → ∞ or the ‘large distance’ limit z → ∞ are relevant to the behaviour of solutions. Since we are interested in eigenfunctions

(1.2)  (Δ + V − λ)u = 0,

for finite λ, only the latter limit is important and hence our microlocal analysis takes place in the vicinity of the finite (fixed) energy, or characteristic, surface which can be written formally

(1.3)  Σ(λ) = {|ζ|^{2} + V (z) = λ, |z| = ∞} .

Setting |z| = ∞ is restriction to the sphere at infinity for the radial compactification \overline{ℝ^{n}} of ℝ^{n} so Σ(λ) ⊂ 𝕊_{ω}^{n−1} × ℝ^{n}, z = |z|ω. This phase space at infinity is a contact manifold where the contact form is

(1.4)  α = −dν + μ · dω, ζ = μ ⊕ νω, μ ∈ ω^{⊥}.

In the general case of a scattering metric on a compact manifold with boundary (see [13]) the corresponding phase space consists of

(1.5)  C_{∂} = ^{sc}T_{Y}^{∗} X,

the restriction of the scattering cotangent bundle to the boundary. Let (x, y) be local coordinates near a boundary point, where x is a local defining function and y restrict to coordinates on Y. Then the forms dx/x^{2} and dy/x are a basis for the fibre ^{sc}T_{p}^{∗}X at each point p in the coordinate chart, and hence one may write an arbitrary point q ∈ ^{sc}T^{∗}X in the form

(1.6)  q = −ν \frac{dx}{x^{2}} + ∑_{i} μ_{i} \frac{dy_{i}}{x} ,

giving local coordinates (x, y, ν, μ) on ^{sc}T^{∗}X. The characteristic variety of P at energy λ then is

(1.7)  Σ(λ) = {|(ν, μ)|_{y}^{2} + V (y) = λ}

where | · |_{y}^{2} is the metric function at the boundary.

Again C_{∂} is naturally a contact manifold. The contact structure arises as the ‘boundary value’ of the singular symplectic structure on ^{sc}T^{∗}X, corresponding to the fact that the associated semiclassical model is the leading part, at the boundary, of the Hamiltonian sytem defined by the energy. In terms of local coordinates as above, the symplectic form, arising from the identification ^{sc}T^{∗}X^{∘} ≃ T^{∗}X^{∘}, is

(1.8)  ω̃ = d(νd\frac{1}{x} + μ · \frac{dy}{x}) = (−dν + μ · dy) ∧ \frac{dx}{x^{2}} + \frac{dμ ∧ dy}{x} ,

and is hence singular at the boundary. Since the vector field x^{2}∂_{x} is well-defined at the boundary, modulo multiples, the contact form

(1.9)  α = ω̃(x^{2}∂_{x}, ·) = −dν + μ · dy

defines a line subbundle of T^{∗}C_{∂} which fixes the contact structure on C_{∂}.

The Hamilton vector field H_{p} of P, regarded as a vector field on ^{sc}T^{∗}X, is of the form xW̃, where W̃ is tangent to ∂(^{sc}T^{∗}X). Here W has the form

W̃ = −2νx∂_{x} + W, W a vector field on ^{sc}T_{∂X}^{∗}X.

On Σ(λ) = {p = λ}, we have x^{−1}H_{p} = x^{−1}H_{p−λ} = H_{x^{−1}(p−λ)} − (p − λ)H_{x^{−1}} = H_{x^{−1}(p−λ)}. Since x^{−1}(p − λ) is order −1, its Hamilton vector field W_{λ} restricted to ∂(^{sc}T^{∗}X) is the contact vector generated by the Hamiltonian p − λ on ∂(^{sc}T^{∗}X). It is determined directly in terms of the contact form by

(1.10)  dα(·, W_{λ}) + γα = dp, α(W_{λ}) = p − λ,

for some function γ. The vector field W_{λ} is tangent to Σ(λ) whenever the latter is smooth, as follows by pairing (1.10) with W_{λ}. By the semiclassical model at energy λ we mean the flow defined by W = W_{λ} on Σ(λ).

Lemma 1.1. For any scattering metric and any real V ∈ C^{∞}(X) the characteristic surface Σ(λ) is smooth whenever λ is not a critical value of V_{0}. For regular λ the critical set of the vector field W on Σ(λ) is the union of the two radial sets (1.11) RP_{±}(λ) = \{(y, ν, μ) ∈ Σ(λ); ν = ±\sqrt{λ − V (p)}, μ = 0, y = p where d_{Y} V (p) = 0\}

and projection onto Y gives bijections

(1.12)  π : RP_{±}(λ) ⟶ Crit(V) ∩ {y ∈ Y; V (y) < λ}

for each sign.

Proof. The vector field is fixed by (1.10) so in any local coordinates y in the boundary it follows from (1.9) that

(1.13)  dμ ∧ dy(·, W) = γ(dν − μdy) + dp.

Given a boundary point q we may choose Riemannian normal coordinates based at q in the boundary. Thus the boundary metric is Euclidean to second order, so p = ν^{2} + |μ|_{y}^{2} + O(|y|^{2}|μ|^{2}) + V (y) − λ and we may easily invert (1.13) to find that at the fibre above q

(1.14)  W = 2μ · ∂_{y} − (V′(y) + 2νμ) · ∂_{μ} + 2|μ|_{y}^{2}∂_{ν} + O(|y||μ|), γ = −2ν.

This can only vanish when μ = 0 and dV_{0}(z) = 0 giving (1.11). □

We remark that in the Euclidean setting, (1.11) amounts to

(1.15)  RP_{±}(λ) = {(y, ζ) ∈ Σ(λ); y = ±ζ/|ζ|, y = z where d_{Y} V (z) = 0}.

If V_{i} = V |_{Y_{i}} is perfect Morse then the component of Σ(λ) above Y_{i},

(1.16)  Σ_{i}(λ) is a \{ \begin{matrix}sphere for & κ_{i} = min(V |_{Y_{i}}) < λ < K_{i} = max(V |_{Y_{i}}), \\ torus for & λ > K_{i}. \end{matrix}

In the general Morse case, Σ_{i}(λ) is a union of disjoint spheres for non-critical λ < K_{i} and a torus for λ > K_{i}.

In all cases if λ is not a critical value of V |_{Y_{i}} then

(1.17)  I_{i}(λ) = Σ_{i}(λ) ∩ {ν = 0}

is a smooth curve (empty if λ < κ_{i}, generally with several components) to which W is transversal. The flow is symmetric under ν → −ν, μ → −μ.

Proposition 1.2. Provided dim X = 2 and V_{0} is Morse the critical point q_{±}(λ) = (z, ±\sqrt{λ − V (z)}, 0) in (1.11) is

(1.18)  \begin{cases}a centre if z is a minimum and λ < λ_{Hess}(z) \\ a sink/source if z is a minimum and λ ≥ λ_{Hess}(z) \\ a saddle if z is a maximum, \end{cases}

where λ_{Hess}(z) = V (z) + 2V′′(z).

Proof. As already noted above, (1.14) is valid locally in geodesic boundary coordinates. If z is a critical point for V then μ = 0 at the critical points z_{±}(λ) of W on Σ(λ) and we may use local coordinates y, μ in the characteristic variety nearby, since ν = ±\sqrt{λ − V (y) − μ^{2}} is smooth. In these coordinates the linearization of W inside Σ(λ) at the critical point is

(1.19)  L = 2[μ∂_{y} − (ay + ν̃μ)∂_{μ}], where V′′(z) = 2a, ν̃ = ±\sqrt{λ − V (z)}.

The eigenvalues of L are therefore the roots of

(1.20)  (\frac{s}{2})^{2} + ν̃(\frac{s}{2}) + a = 0.

It is convenient for future reference to write the eigenvalues s_{j} as

(1.21)  s_{j} = −2ν̃r_{j};

the r_{j} thus satisfy

(1.22)  r^{2} − r + \frac{a}{ν̃^{2}} = 0.

If a < 0, so z is a local maximum then the eigenvalues of L are real, and are given by

(1.23)  r_{1} = \frac{1}{2} − \sqrt{\frac{1}{4} − \frac{a}{ν̃^{2}}} < 0 < 1 < r_{2} = \frac{1}{2} + \sqrt{\frac{1}{4} − \frac{a}{ν̃^{2}}} ,

so the critical point is a saddle. If a > 0, so z is a local minimum, then the discriminant

(1.24)  1 − \frac{4a}{ν̃^{2}} is \{ \begin{matrix}negative & if λ < λ_{Hess} \\ zero & if λ = λ_{Hess} \\ positive & if λ > λ_{Hess} \end{matrix} , λ_{Hess} = V (z) + 2V′′(z).

Correspondingly the eigenvalues are of the form −2ν̃r_{j}, j = 1, 2, where, respectively,

\begin{matrix}r_{1} = \frac{1}{2} + i\sqrt{\frac{a}{ν̃^{2}} − \frac{1}{4}}, r_{2} = \frac{1}{2} − i\sqrt{\frac{a}{ν̃^{2}} − \frac{1}{4}}, \\ r_{1} = r_{2} = 1/2, \\ 0 < r_{1} = \frac{1}{2} − \sqrt{\frac{1}{4} − \frac{a}{ν̃^{2}}} < \frac{1}{2} < r_{2} = \frac{1}{2} + \sqrt{\frac{1}{4} − \frac{a}{ν̃^{2}}} < 1 \end{matrix} ; (1.25)

and the critical point is a centre, a degenerate center, or a source/sink. □

Remark 1.3. If λ ≠ λ_{Hess}(z), a basis for the eigenvectors of L is

(1.26)  e_{j} = (s_{j}/2 + ν̃) dy + dμ = ν̃(1 − r_{j}) dy + dμ.

If λ = λ_{Hess}, L has only one eigenvector, \frac{1}{2} ν̃ dy +dμ, but the generalized eigenspace (of eigenvalue −ν̃) is of course two-dimensional.

We also need a result about integral curves associated to the eigenvectors at a sink.

Proposition 1.4. Suppose that q ∈ Min_{±}(λ), and let e_{j} be the eigenvectors of L as in (1.26). If r_{2}/r_{1} > 1 is not an integer, then there are two smooth one-dimensional submanifolds L_{j} through z such that W is tangent to L_{j} in a neighbourhood of q and e_{j} ∈ N^{∗}L_{j}. If r_{2}/r_{1} = N is an integer, then there is one smooth Legendre curve L_{1} as above and another smooth curve, L_{2}, such that W = W_{t} + aW̃ , with W_{t} tangent to L_{2}, and where a vanishes at z to order N i.e. a ∈ I^{N}, I being the ideal of functions vanishing at z.

Proof. By a linear change of variables, (y, μ) → (v_{1}, v_{2}), we may assume that W takes the form

W = r_{1}v_{1} \frac{∂}{∂v_{1}} + r_{2}v_{2} \frac{∂}{∂v_{2}} + O(|v|^{2})

near v = 0. In these coordinates, the eigenvector e_{i} corresponds to dv_{i}. Suppose first that r_{2}/r_{1} is not an integer. Then the eigenvalues are nonresonant, and are both positive, so by the Sternberg linearization theorem [16] there is a smooth change of variables so that W takes the form

W = r_{1}v_{1} \frac{∂}{∂v_{1}} + r_{2}v_{2} \frac{∂}{∂v_{2}} .

Then we may take L_{i} = {v_{i} = 0}. In the resonant case, r_{2}/r_{1} = N ≥ 2, we still obtain a normal form, but we are unable to remove resonant terms on the right hand side. In this case, there is just one term, so we find [16] that there is a smooth change of variables so that W takes the form

W = \frac{1}{N + 1}v_{1} \frac{∂}{∂v_{1}} + \frac{N}{N + 1}v_{2} \frac{∂}{∂v_{2}} + cv_{1}^{N} \frac{∂}{∂v_{2}} .

Again, if we take L_{i} = {v_{i} = 0} then we satisfy the conditions of the theorem. □

The corresponding result at saddle points is the stable/unstable manifold theorem.

Proposition 1.5. Suppose that q = z_{±}(λ) is a saddle, and let e_{j} be the eigenvectors of L as in (1.26). Then there are two smooth Legendre curves L_{j} through z such that W is tangent to L_{j} and e_{j} ∈ N^{∗}L_{j}. Moreover, if ν(q) < 0, then ν ≥ ν(q) on L_{1} and ν ≤ ν(q) on L_{2}, while if ν(q) > 0, then ν ≤ ν(q) on L_{1} and ν ≥ ν(q) on L_{2}.

Proof. The first part follows from the stable/unstable manifold theorem, while the second part follows from (1.27). □

Remark 1.6. In both cases, there are coordinates (v_{1}, v_{2}) on Σ(λ) near q such that W restricts to r_{2}v_{2}∂_{v_{2}} on L_{1} and r_{1}v_{1}∂_{v_{1}} on L_{2} near q.

Below we make use of open neighbourhoods of the critical points which are wellbehaved in terms of W.

Definition 1.7. By a W -balanced neighbourhood of a critical point q ∈ Σ(λ), λ ∉ Cv(V), we shall mean an open neighbourhood, O of q in C_{∂} which contains no other radial point, which meets Σ(λ) in a W -convex set (that is, each integral curve of W meets Σ(λ) in a single interval, possibly empty) and is such that the closure of each integral curve of W in O meets ν = ν(q).

Lemma 1.8. Any critical point q ∈ RP(λ) for λ > V (π(q)) has a neighbourhood basis of W -balanced open sets.

Proof. In the case of sinks or sources this is clear, since arbitrarily small balls around these that are convex with respect to W are W -balanced. In the case of saddle points we may use the stable manifold theorem. It suffices to suppose that ν(q) > 0. Taking a neigbourhood of L_{2} ∩ {ν = ν(q) − ϵ}, for ϵ > 0 but small, a W -balanced neighbourhood is given by the W -flow-out of this set in the direction of increasing ν strictly between ν = ν(q) − ϵ and ν = ν(q) + ϵ, together with the parts of L_{1} and L_{2} in |ν − ν(q)| < ϵ. □

Consider the global structure of the dynamics of W. From (1.14) it follows directly that

(1.27) W (ν) = 2|μ|_{y}^{2} ≥ 0, i.e. ν is non-decreasing along integral curves of W.

In fact, along maximally extended integral curves of W, ν is only constant if the curve reduces to a critical point. It is also easy to see that W has no non-trivial periodic orbits and every maximally extended bicharacteristic γ : ℝ_{t} → Σ(λ) tends to a point in RP_{+}(λ) ∪ RP_{−}(λ) as t → ±∞. Indeed, lim_{t→±∞} ν(γ(t)) = ν_{±} exists by the monotonicity of ν, and any sequence γ_{k} : [0, 1] → Σ(λ), γ_{k}(t) = γ(t_{k} + t), t_{k} → +∞, has a uniformly convergent subsequence, which is then an integral curve γ̃ of W. Then ν is constant along this bicharacteristic, hence h is identically 0. In view of the ∂_{μ} component of W in (1.14), ∂_{y}V is identically 0 along γ̃, hence y is a critical point of V and the limit is a point in RP_{+}(λ) ∪ RP_{−}(λ).

We can now define the forward, resp. backward, bicharacteristic relation as follows.

Definition 1.9. The forward bicharacteristic relation Φ_{+} ⊂ Σ(λ) × Σ(λ) is the set of (ξ′, ξ) for which there exist bicharacteristics γ_{j}, j = 1, …, N, N ≥ 1, and T_{−}, T_{+} ∈ ℝ, T_{−} ≤ T_{+}, such that γ_{1}(T_{−}) = ξ′, γ_{N}(T_{+}) = ξ, and lim_{t→+∞} γ_{j}(t) = lim_{t→−∞} γ_{j+1}(t) for j = 1, 2, …, N − 1.

The backward bicharacteristic relation Φ_{−} is defined similarly, with the role of ξ and ξ′ reversed. The relations Φ_{±} depend on λ, but this is not indicated in notation.

In view of the previous observations, lim_{t→±∞} γ_{j}(t) ∈ RP_{+}(λ) ∪ RP_{−}(λ), so the content of the definition is to allow ξ and ξ′ to be connected by a chain of bicharacteristics in the forward direction. Note that Φ_{+} is reflexive and transitive, but not symmetric.

Also, if ξ, ξ′ ∈ RP_{+}(λ) ∪ RP_{−}(λ), and there exists a bicharacteristic γ such that lim_{t→+∞} γ(t) = ξ, lim_{t→−∞} γ(t) = ξ′, then (ξ′, ξ) ∈ Φ_{+}. Indeed, we can take γ_{1} and γ_{3} to be appropriately parameterized constant bicharacteristics with image ξ′, resp. ξ, and γ_{2} = γ. Of course, γ may be replaced by a chain of bicharacteristics.

We recall that the image of a set K ⊂ ^{sc}T_{Y}^{∗} X under Φ_{+} is the set

(1.28)  Φ_{+}(K) = {ξ ∈ ^{sc}T_{Y}^{∗} X; ∃ ξ′ ∈ K s.t. (ξ′, ξ) ∈ Φ_{+}}.

We will be interested in the forward flow-out Φ_{+}(RP_{+}(λ)) of the outgoing radial set. Since ν > 0 on RP_{+}(λ), and is increasing along bicharacteristics, ν > 0 on Φ_{+}(RP_{+}(λ)) as well.

Lemma 1.10. If q ∈ RP_{+}(λ) is a center or sink, then Φ_{+}({q}) = {q}. On the other hand, if q is a saddle, then with the notation of Proposition 1.5, Φ_{+}({q}) is locally (in Σ(λ)) given by the local unstable curve L_{2}, i.e. by the unique smooth Legendre manifold L_{2} such that ν ≥ ν(q) on L_{2}, q ∈ L_{2} and W is tangent to L_{2}.

Remark 1.11. Suppose that λ ∉ Cv(V). Then on Φ_{+}(RP_{+}(λ)), ν ≥ a(λ), and on Φ_{−}(RP_{−}(λ)), ν ≤ −a(λ), where a(λ) = min{ν(q); q ∈ RP_{+}(λ)} > 0.

2. Blow-ups and resolution of singularities of flows

As indicated in the Introduction, to aid in the description of the asymptotic behaviour of eigenfunctions it is very convenient to modify the underlying manifold with boundary by blowing it up at the minima of V_{0} and with homogeneity r with respect to the boundary. We give a brief discussion of the blow-up of a stable critical point of a vector field with respect to the vector field.

Suppose that W is a real C^{∞} vector field on a manifold M, and that o ∈ M, with W (o) = 0, is a linearly stable critical point. That is, the eigenvalues of the linearization of W at o all have negative real parts. We handle unstable critical points by changing the sign of the vector field.

If Φ : M ×ℝ_{t} → M denotes the flow generated by W, then o has a neighbourhood O such that for p ∈ O, lim_{t→+∞} Φ(p, t) = o. There always exists a closed embedded submanifold, diffeomorphic to a sphere, S ⊂ O which is transversal to W. It also necessarily satisfies

(2.1)  lim_{t→+∞} Φ(S, t) = {o}.

Then Φ restricts to a diffeomorphism of S × [0, +∞) to O′ \ {o} where O′ is the union of S and the neighbourhood of o consisting of one of the components of M \S.

We may compactify S × [0, +∞)_{t} by embedding it as a dense subset

S × [0, +∞)_{t} ↪ S × [0, 1], (s, t) ⟼ (s, e^{−t}).

This makes e^{−t} a defining function for a boundary hypsersurface. Using the diffeomorphism Φ this compactifies O′ \ {o} to a manifold with boundary. Finally then, replacing O′ as a subset of M by S × [0, 1] using Φ to identify O′ \ {o} with S ×(0, 1) we obtain a compactification of M \{o} which is a compact manifold with boundary.

Lemma 2.1. The compact manifold with boundary obtained by blow-up, with respect to a vector field W, of a linearly stable critical point, {o}, is independent of the transversal S to W, satisfying (2.1), used to define it.

We may therefore denote the blown up manifold by [M, {o}]_{W} .

Proof. Since there are spheres transversal to W in any neighbourhood of {o} it suffices to show that the two manifolds obtained using a transversal S and a second transversal S̃ ⊂ O′ are naturally diffeomorphic where O′ is the component of M \ S containing o. Since it lies within S, S′ is precisely the image of Φ on {(s, T (s)); s ∈ S} for some smooth map T : S ⟶ (0, ∞). In terms of the compactification this replaces e^{−t} by e^{−t}e^{−T (s)} which induces a diffeomorphism of the neighbourhoods of the boundaries of the two compactifications. □

Thus the abstract manifold, [M, {o}]_{W} , defined in this way is, as a set, the union of M \{o} with an abstract sphere as boundary. The boundary can be realized more concretely by using the flow of W to identify any two transversals to W which are sufficiently close to {o}. The Lemma above shows that [M, {o}]_{W} has a natural C^{∞} structure as a compact manifold with boundary with interior which is canonically diffeomorphic to M \ {o}. The boundary hypersurface is called the front face of the blow-up and is usually denoted below as ff, or more precisely as ff([M; {o}]_{W}). Note that it is not generally the case that the natural map

(2.2)  [M, {o}]_{W} ⟶ M,

under which ff ⟼ {o}, is smooth.

More generally, if X is a smooth manifold with boundary and W is a smooth vector field on X which is tangent to the boundary then we may use the construction above to define [X, {o}]_{W} even for a stable critical point of W on the boundary. Namely, we may simply extend W across the boundary to the double of X and then observe that the closure of the preimage of X \ {o} in [2X, {o}]W̃ is independent of the extension W̃ of W.

One particular use of this construction is to define inhomogeneous blow-ups. Let X be a 2-dimensional manifold with boundary, with o ∈ ∂X. Choose local coordinates (x, y) near o such that x ≥ 0 is a boundary defining function and o is given by x = 0, y = 0. Let r ∈ (0, 1) be a given homogeneity. We wish to blow up o in X so that y is homogeneous of degree 1 and x is homogeneous of degree 1/r. To do so consider the vector field −W = r^{−1}x∂_{x} + y∂_{y}, and apply the construction above, denoting the result [X, {o}]_{r}; clearly it does not depend on how W is extended outside a neighbourhood of o. Observe that y/x^{r} and x/|y|^{1/r} are homogeneous of degree 0, i.e. are annihilated by W, where they are bounded, so they can be regarded as variables on the transversal S. Thus, local coordinates on the blown-up space, in the lift of the region |y/x^{r}| < C are given by y/x^{r} and x^{r}, while local coordinates in the lift of the region x/|y|^{1/r} < C′ are given by x/|y|^{1/r} and y.

In fact the manifold [X; {o}]_{r} does depend on the choice of W, and so on the choice of coordinates (x, y). However the space of classical conormal functions, with respect to the boundary, is defined independently of choices. This is in essence because x has homogeneity greater than that of the boundary variable. More explicitly, any change of coordinates takes the form y′ = a(x, y)y + b(x, y)x, where x′ = c(x, y)x, c(0, 0) > 0 and a(0, 0) ≠ 0. So, for example, y′/(x′)^{r} = (a/c^{r})(y/x^{r})+ (b/c^{r})x^{1−r} which is bounded (in x ≤ x_{0}, x_{0} > 0, |y| ≤ y_{0}, y_{0} > 0) if and only if y/x^{r} is bounded (since r ∈ (0, 1)). Such calculations show that the blow-ups using (x, y) and using (x′, y′) coincide as topological manifolds. Their C^{∞} structures are different as can be seen from the appearance of terms involving x^{1−r} in the transformation law. However, it is easy to see that the class of conormal functions, and also that of polyhomogeneous conormal functions, on [X; {o}]_{r}, is well-defined, independent of the choice of local coordinates used in the definition (although the orders may need to be appropriately adjusted). Thus, [X; {o}]_{r} may be thought of as a ‘conormal manifold’ rather than as a C^{∞} manifold, i.e. the algebra of smooth functions should be replaced by the algebra of polyhomogeneous conormal functions as the basic object of interest.

Such inhomogeneous blow-ups may be generalized to higher dimensions, although there one needs extra structure to make it independent of the choice of coordinates. Since we do not need this notion in the present paper, we do not pursue it further.

3. Limiting absorption principle

In later sections we prove, by positive commutator methods, various results of propagation of singularities type. Here we summarize, in four theorems, the main conclusions concerning the limit of the resolvent on the real axis and the behaviour of L^{2} eigenfunctions. The scattering wavefront set is discussed in [13] as are the scattering Sobolev spaces H_{sc}^{m,r}(X); the characteristic variety Σ(λ), scattering relation Φ_{+} and radial sets RP_{±}(λ) are defined in Section 1 above.

First, we recall Hörmander’s theorem on the propagation of singularities; in the present setting this is [13, Proposition 7].

Theorem 3.1 (Propagation of singularities). For u ∈ C^{−∞}(X), and any real m and r,

WF_{sc}^{m,r−1}(u) \ WF_{sc}^{m,r} ((P − λ)u) is a union of maximally extended
bicharacteristics of P − λ in Σ(λ) \ WF_{sc}^{m,r} ((P − λ)u).

Similarly, if u_{t}, for t ∈ G ⊂ (0, 1] is a bounded family in H_{sc}^{M,R}(X), then

WF_{sc,L^{∞}(G_{t})}^{m,r−1}(u_{t}) \ WF_{sc,L^{∞}(G_{t})}^{m,r} ((P − (λ + it))u_{t}) is a union of maximally

forward-extended bicharacteristics of P − λ in Σ(λ) \ WF_{sc,L^{∞}(G_{t})}^{m,r} ((P − λ)u) .

Theorem 3.2 (Decay of outgoing eigenfunctions). If λ ∉ Cv(V) then the space E_{pp}(λ) is finite dimensional, is contained in Ċ^{∞}(X), and may be characterized by

(3.1) E_{pp}(λ) = {u ∈ C^{−∞}(X); (P − λ)u = 0 and

WF_{sc}^{m,l}(u) ∩ RP_{+}(λ) = ∅ for some l > −1/2}.

The set of eigenvalues σ_{pp}(P) = {λ; E_{pp}(λ) ≠ {0}} is discrete outside Cv(V) and is contained in (− inf_{X} V, sup Cv(V)].

As an immediate consequence of Theorem 3.2,

(3.2)  E_{ess}^{−∞}(λ) = {u ∈ C^{−∞}(X); (P − λ)u = 0, ⟨u, v⟩ = 0 ∀ v ∈ E_{pp}(λ)}

is well defined for λ ∉ Cv(V) as are spaces such as

H_{sc}^{m,r}(X) ⊖ E_{pp}(λ) = {f ∈ H_{sc}^{m,r}(X); ⟨f, ϕ⟩ = 0 ∀ ϕ ∈ E_{pp}(λ)}.

Theorem 3.3 (Limiting absorption principle). The resolvent

(3.3)  R(λ + it) = (P − (λ + it))^{−1}, 0 ≠ t ∈ ℝ, λ ∉ Cv(V)

extends continuously to the real axis, i.e. R(λ ± i0) exist, as bounded operators

(3.4) H_{sc}^{m,r}(X) ⊖ E_{pp}(λ) ⟶ H_{sc}^{m+2,l}(X) ⊖ E_{pp}(λ), ∀ m ∈ ℝ, r > 1/2, l < −1/2.

Theorem 3.4 (Forward propagation). For λ ∉ Cv(V) and f ∈ Ċ^{∞}(X) ⊖ E_{pp}(λ),

(3.5)  WF_{sc}(R(λ + i0)f) ⊂ Φ_{+}(RP_{+}(λ)) ⊂ {ν ≥ 0}

and R(λ + i0) extends by continuity to

(3.6) R(λ + i0) : {v ∈ C^{−∞}(X) ⊖ E_{pp}(λ); WF_{sc}(v) ∩ Φ_{−}(RP_{−}(λ)) = ∅} ⟶

C^{−∞}(X),

with wavefront set bound given by

(3.7) WF_{sc}(R(λ + i0)v) ⊂ WF_{sc}(v) ∪ Φ_{+}(WF_{sc}(v) ∩ Σ(λ)) ∪ Φ_{+}(RP_{+}(λ)).

We next show that various statements in the Introduction are direct consequences of these results.

Proposition 3.5. For λ ∉ Cv(V) the space (P −λ)Ċ^{∞}(X) is closed in Ċ^{∞}(X) and for λ ∈ σ_{pp}(P) \ Cv(V), (P − λ)Ċ^{∞}(X) is L^{2}-orthogonal to E_{pp}(λ) and

(3.8)  R(λ + i0)f = R(λ − i0)f ∀ f ∈ (P − λ)Ċ^{∞}(X).

Proof. By assumption λ ∉ Cv(V) so E_{pp}(λ) ⊂ Ċ^{∞}(X). The self-adjointness of P gives

⟨(P − λ)v, ϕ⟩ = ⟨v, (P − λ)ϕ⟩ = 0 if ϕ ∈ E_{pp}(λ)

so (P − λ)Ċ^{∞}(X) is L^{2} orthogonal to E_{pp}(λ).

Suppose u is in the closure of (P −λ)Ċ^{∞}(X) in Ċ^{∞}(X). Thus u_{j} = (P −λ)v_{j} → u with v_{j} ∈ Ċ^{∞}(X). By the finite dimensionality of E_{pp}(λ) we may assume that v_{j} ⊥ E_{pp}(λ). Now consider w_{j} = R(λ + i0)u_{j} which exist by Theorem 3.3 and by Theorem 3.4 satisfy WF_{sc}(w_{j}) ⊂ Φ_{+}(RP_{+}(λ)). Thus,

(P − λ)(v_{j} − w_{j}) = 0 and WF_{sc}(v_{j} − w_{j}) ⊂ Φ_{+}(RP_{+}(λ))

so, by Theorem 3.2, v_{j} − w_{j} ∈ E_{pp}(λ), to which both terms are orthogonal, so v_{j} = w_{j} = R(λ + i0)u_{j}. Since u_{j} → u in Ċ^{∞}(X), v_{j} → R(λ + i0)u in C^{−∞}(X). The same argument shows that v_{j} → R(λ − i0)u in C^{−∞}(X), so v = R(λ + i0)u = R(λ − i0)u. By (3.5) applied with each sign, WF_{sc}(v) is disjoint from both {ν ≤ 0} and {ν ≥ 0} and hence is empty. Thus v ∈ Ċ^{∞}(X), and since u = (P − λ)v, (P − λ)Ċ^{∞}(X) is closed in Ċ^{∞}(X); this also proves (3.8). □

Corollary 3.6. For λ ∉ Cv(V), the space Ċ^{∞}(X)/(P − λ)Ċ^{∞}(X) is a Fréchet space, as is (Ċ^{∞}(X) ⊖ E_{pp}(λ))/(P − λ)Ċ^{∞}(X).

An abstract parameterization of generalized eigenfunctions, which has little specific to do with our particular problem, may be obtained via the two terms of (I.12). Namely, let

(3.9)  RR_{ess,±}^{∞}(λ) = [R(λ ± i0)(Ċ^{∞}(X) ⊖ E_{pp}(λ))]/(Ċ^{∞}(X) ⊖ E_{pp}(λ))

be the range of the incoming (−), resp. outgoing (+), boundary value of the resolvent acting on Schwartz functions, modulo Schwartz functions. It is now immediate that the map

(P − λ) : RR_{ess,±}^{∞}(λ) ⟶ (Ċ^{∞}(X) ⊖ E_{pp}(λ))/(P − λ)Ċ^{∞}(X)

is an isomorphism (for each sign). This induces a Fréchet topology on RR_{ess,±}^{∞}(λ).

Proposition 3.7. For λ ∉ Cv(V) ∪ σ_{pp}(P) the nullspace of

Sp(λ) = \frac{1}{2πi} (R(λ + i0) − R(λ − i0)) : Ċ^{∞}(X) ⟶ E_{ess}^{∞}(λ)

is (P − λ)Ċ^{∞}(X); if λ ∈ σ_{pp}(P) \ Cv(V), the same remains true for the map (2πi)^{−1} (R(λ + i0) − R(λ − i0)) : Ċ^{∞}(X) ⊖ E_{pp}(λ) → E_{ess}^{∞}(λ), for λ ∈ σ_{pp}(P) \ Cv(V).

Proof. By (3.8), (P − λ)Ċ^{∞}(X) lies in the nullspace of Sp(λ). Conversely, suppose that f ∈ Ċ^{∞}(X), Sp(λ)f = 0, so v = R(λ + i0)f = R(λ − i0)f. Since WF_{sc}(R(λ + i0)(f)) and WF_{sc}(R(λ − i0)(f)) are disjoint (ν < 0 on one, ν > 0 on the other), both of these are empty, so v ∈ Ċ^{∞}(X), and (P − λ)v = f is in (P − λ)Ċ^{∞}(X) as claimed. □

Corollary 3.8. If λ ∉ Cv(V) ∪ σ_{pp}(P) the spaces E_{ess}^{∞}(λ), RR_{ess,+}^{∞}(λ), RR_{ess,−}^{∞}(λ) and Ċ^{∞}(X)/(P − λ)Ċ^{∞}(X) are all isomorphic; this remains true for λ ∈ σ_{pp}(P) \ Cv(V) with Ċ^{∞}(X)/(P − λ)Ċ^{∞}(X) replaced by (Ċ^{∞}(X) ⊖ E_{pp}(λ))/(P − λ)Ċ^{∞}(X).

Explicitly, the isomorphisms

(3.10)  I_{±}(λ) : E_{ess}^{∞}(λ) → RR_{ess,±}^{∞}(λ)

are the maps

\begin{matrix}I_{+}(λ) : u = [R(λ + i0) − R(λ − i0)]f ↦ R(λ + i0)f, \\ I_{−}(λ) : u = [R(λ + i0) − R(λ − i0)]f ↦ −R(λ − i0)f, f ∈ Ċ^{∞}(X). \end{matrix}

Although f is not well-defined in Ċ^{∞}(X), it is in Ċ^{∞}(X)/(P − λ)Ċ^{∞}(X), hence the image of R(λ + i0)f in RR_{ess,+}^{∞}(λ) is also well defined. By Theorem 3.4,

WF_{sc}(I_{±}(λ)u) ⊂ Φ_{±}(RP_{±}(λ)).

Thus, I_{−}(λ) maps u ∈ E_{ess}^{∞}(λ) to its ‘incoming part’, while I_{+}(λ) maps u to its outgoing part. We define the abstract scattering matrix as the map

(3.11)  S(λ) : RR_{ess,−}^{∞}(λ) → RR_{ess,+}^{∞}(λ), S(λ) = I_{+}(λ)I_{−}(λ)^{−1}.

In Section 11 we identify RR_{ess,±}^{∞}(λ) geometrically. We use this identification in Section 12 to extend S(λ) to a unitary operator on a Hilbert space, giving rise to the definition of the S-matrix in the introduction in (I.36).

We now turn to the relationship between E_{ess}^{∞}(λ) and E_{ess}^{−∞}(λ) which has two facets. On the one hand, as we show below, E_{ess}^{∞}(λ) is dense in E_{ess}^{−∞}(λ) in the topology of C^{−∞}(X), indeed in a stronger sense. On the other hand they are dual to each other with respect to a form of ‘Green’s pairing’ that we describe in Section 12.

By Theorem 3.4, Sp(λ) may be applied to any distribution which is L^{2} orthogonal to E_{pp}(λ), provided its scattering wavefront set is contained in {|ν| < ϵ} for ϵ > 0 sufficiently small, namely ϵ < a(λ),

a(λ) = min{ν(q); q ∈ RP_{+}(λ)} > 0.

Proposition 3.9. If λ ∉ Cv(V) and ϵ > 0 then every u ∈ E_{ess}^{−∞}(λ) is of the form Sp(λ)f for some f ∈ C^{−∞}(X) with WF_{sc}(f) ⊂ {|ν| < ϵ} and f ⊥ E_{pp}(λ).

Of course if λ ∉ σ_{pp}(P) the orthogonality condition is void.

Proof. Suppose first that λ ∉ σ_{pp}(P).

Let A ∈ Ψ_{sc}^{−∞,0}(X) be a pseudodifferential operator with boundary symbol given by ψ_{1}(σ_{∂}(P − λ))ψ_{2}(ν/a(λ)), where ψ_{1}(t) is supported in |t| < ϵ and ψ_{2}(t) is supported in t > 1/4, and equal to 1 when t > 1/2. Then, by Theorem 3.4, we may apply R(λ − i0) to (P − λ)(Id −A)u and R(λ + i0) to (P − λ)Au. Since

(P − λ)((Id −A)u − R(λ − i0)(P − λ)(Id −A)u) = 0,

and this function has scattering wavefront contained in ν ≤ a(λ)/2, we conclude using Theorem 3.2 that (Id −A)u = R(λ − i0)(P − λ)(Id −A)u. Similarly, Au = R(λ + i0)(P − λ)Au. Since (P − λ)Au = [P, A]u = −(P − λ)(Id −A)u, we may take f = (2πi)[P, A]u.

If λ ∈ σ_{pp}(P), only a minor modification is necessary. Namely, we consider ΠA in place of A where Π is orthogonal projection off E_{pp}(λ). Note that Id −Π ∈ Ψ_{sc}^{−∞,∞}(X). Then the rest of the argument goes through. □

In fact, this proof shows more.

Proposition 3.10. Suppose λ ∉ Cv(V) and that A ∈ Ψ_{sc}^{−∞,0}(X) has

WF′_{sc}(A) ⊂ {ν > −a(λ)} and WF′_{sc}(Id −A) ⊂ {ν < a(λ)}

where 0 < a(λ) = min{ν(q); q ∈ RP_{+}(λ)}; if Π is the orthogonal projection off E_{pp}(λ) then

(3.12)  2πi Sp(λ)Π[P, A] : C^{−∞}(X) ⟶ C^{−∞}(X)

is continuous, with range in E_{ess}^{−∞}(λ) and the restriction of (3.12) to E_{ess}^{−∞}(λ) is the identity map.

Proof. By assumption −a(λ) < ν < a(λ) on WF′_{sc}(A) ∩ WF′_{sc}(Id −A) so this is also holds on WF′_{sc}([P, A]). The continuity of (3.12) follows. The final statement is a consequence of the proof of the preceeding proposition, namely that, given u ∈ E_{ess}^{−∞}(λ) one can take f = 2πiΠ[P, A]u and conclude that u = Sp(λ)f. □

This result allows us to characterize the topology on E_{ess}^{∞}(λ) three different ways. Let T_{1} be the topology on E_{ess}^{∞}(λ) as in Corollary 3.8, induced by Sp(λ) : (Ċ^{∞}(X)⊖ E_{pp}(λ)) → E_{ess}^{∞}(λ). Fix any A as in Proposition 3.10 and let T_{2} be the topology on E_{ess}^{∞}(λ) induced by the topology of Ċ^{∞}(X) on E_{ess}^{∞}(λ) via the map [P, A] : E_{ess}^{∞}(λ) → Ċ^{∞}(X). Finally let r < −1/2, and let B ∈ Ψ_{sc}^{−∞,0}(X) be such that

(3.13)  B is elliptic at {ν = 0} ∩ Σ(λ), with WF′_{sc}(B) ⊂ {|ν| < a(λ)},

and let T_{3} be the weakest topology on E_{ess}^{∞}(λ) stronger than both the x^{r}L_{sc}^{2}(X) topology on E_{ess}^{∞}(λ) and the topology induced by the map B : E_{ess}^{∞}(λ) → Ċ^{∞}(X).

Proposition 3.11. The topologies T_{1}, T_{2}, T_{3} are equivalent. In particular, T_{2} and T_{3} are independent of choices.

Proof. The equivalence of T_{1} and T_{2} follows from 2πi Sp(λ)Π[P, A] = Id on E_{ess}^{∞}(λ). By the propagation of singularities, T_{3} is stronger that T_{2}. Finally, by the forward propagation, T_{1} is stronger than T_{3}. □

We can also topologize

(3.14)  E_{ess}^{s}(λ) = {u ∈ E_{ess}^{−∞}(λ); WF_{sc}^{0,s−1/2}(u) ∩ {ν = 0} = ∅}, s ∈ ℝ,

similarly. Note first that

E_{ess}^{s}(λ) ⊂ x^{r}L_{sc}^{2}(X) for r < min(−\frac{1}{2}, s − \frac{1}{2}).

For s ∈ ℝ, we let T_{2}^{s} be the topology on E_{ess}^{s}(λ) induced by [P, A] : E_{ess}^{s}(λ) → x^{s+1/2}L_{sc}^{2}(X), so

∥u∥_{T_{2}^{s}} = ∥u∥_{2} = ∥[P, A]u∥_{x^{s+1/2}L_{sc}^{2}(X)}.

In the same setting, let T_{3}^{s} be the topology on E_{ess}^{s}(λ) induced by the x^{r}L_{sc}^{2}(X) topology on E_{ess}^{s}(λ), for r with r < −1/2 and r < s − 1/2, and by the operator B from (3.13), that is by the norm

∥u∥_{T_{3}^{s}} = ∥u∥_{3} = (∥u∥_{x^{r}L_{sc}^{2}(X)}^{2} + ∥Bu∥_{x^{s−1/2}L_{sc}^{2}(X)}^{2})^{1/2} .

For s > 0, we deduce the following as above.

Proposition 3.12. For each s > 0, the topologies T_{2}^{s} and T_{3}^{s} are equivalent and are independent of all choices, making E_{ess}^{s}(λ) into a Hilbert space.

The inclusion map E_{ess}^{s}(λ) ↪ E_{ess}^{s′}(λ), with norm ∥.∥_{3}, is bounded for s′ ≤ s.

We consider s = 0 when we study Green’s pairing in Section 12.

From the density of Ċ^{∞}(X) in the subspace of C^{−∞}(X) with wave front set in {|ν| < ϵ} ∩ Σ(λ), with the topology of Hörmander, and the continuity of Sp(λ) from this space to C^{−∞}(X) we conclude (using Proposition 3.9) that the smooth eigenfunctions, E_{ess}^{∞}(λ), are dense in E_{ess}^{−∞}(λ).

More explicitly, this can be seen by considering u_{r} = ϕ(x/r)u, r > 0 where ϕ ∈ C^{∞}(ℝ) is identically 1 on [1, +∞) and 0 on [0, 1/2). Then Sp(λ)Π[P, A]u_{r} ∈ E_{ess}^{∞}(λ) for r > 0, and u_{r} → u in C^{−∞}(X), hence

E_{ess}^{∞}(λ) ∋ 2πi Sp(λ)Π[P, A]u_{r} → u

in C^{−∞}(X). If u ∈ E_{ess}^{s}(λ), s > 0, then [P, A]u_{r} → [P, A]u in x^{s+1/2}L_{sc}^{2}(X), hence

[P, A]2πi Sp(λ)Π[P, A]u_{r} → [P, A]2πi Sp(λ)Π[P, A]u = [P, A]u

in x^{s+1/2}L_{sc}^{2}(X). More generally, suppose that u ∈ E_{ess}^{s}(λ), s ∈ ℝ, and let Q ∈ Ψ_{sc}^{0,0}(X) be such that WF′_{sc}(Q) ⊂ {|ν| < a(λ)}, WF′_{sc}(Id −Q) ∩ WF′_{sc}([P, A]) = ∅. Let p be such that p < s − 1/2, p < −1/2. Then [P, A]u_{r} → [P, A]u in x^{s+1/2}L_{sc}^{2}(X) shows that 2πi Sp(λ)ΠQ[P, A]u_{r} → 2πi Sp(λ)ΠQ[P, A]u in x^{p}L_{sc}^{2}(X) and (Id −Q)[P, A]u_{r} → (Id −Q)[P, A]u in Ċ^{∞}(X) so 2πi Sp(λ)Π[P, A]u_{r} → u in x^{p}L_{sc}^{2}(X), and, with B from (3.13), 2πiB Sp(λ)Π[P, A]u_{r} → Bu in x^{s−1/2}L_{sc}^{2}(X). We have thus proved the following.

Corollary 3.13. For each λ ∉ Cv(V), E_{ess}^{∞}(λ) is dense in E_{ess}^{−∞}(λ) in the topology of C^{−∞}(X). Moreover, for each λ ∉ Cv(V), s ∈ ℝ, E_{ess}^{∞}(λ) is dense in E_{ess}^{s}(λ) in the topology T_{3}^{s}.

4. Propagation of singularities

In this section, we derive wavefront set bounds on eigenfunctions, or more generally of solutions of (P − λ)u = f, and use these to prove the results of the previous section. Let us first consider λ ∈ ℂ \ [inf V_{0}, +∞). Then the symbol p − λ of σ_{∂}(P − λ) never vanishes, so P − λ has a parametrix in the scattering calculus, i.e. there is G(λ) ∈ Ψ_{sc}^{−2,0}(X) such that E(λ) = (P − λ)G(λ) − Id ∈ Ψ_{sc}^{−1,1}(X). Thus E(λ) is compact on L^{2}. By analytic Fredholm theory we conclude that P has discrete spectrum in (−∞, inf V_{0}). Also ellipticity of p − λ at the boundary implies that WF_{sc}(u) is empty if (P − λ)u = 0, so u ∈ Ċ^{∞}(X).

In the rest of the section we consider λ > inf V_{0}, and use positive commutator estimates to derive wavefront set bounds. First we recall some standard results in the scattering calculus. Since P ∈ Ψ_{sc}^{∗,0}(X), the Hamilton vector field H_{p} of P vanishes to first order at x = 0, we define the scattering Hamilton vector field scH_{p} = x^{−1}H_{p}, which is smooth and nonvanishing up to the boundary. In fact, scH_{p} = W − 2νx∂_{x} in local coordinates near the boundary, where W is the contact vector field from Section 1. If A ∈ Ψ_{sc}^{∗,−l−1}(X) is a scattering pseudodifferential operator, with symbol x^{−l−1}a, then the symbol of the commutator i[P, A] is

(4.1)  σ_{∂}(i[P, A]) = H_{p}(x^{−l−1}a) = x^{−l}(W a + 2(l + 1)νa).

We also recall the notion of wavefront set with respect to a family of functions, WF_{sc,L^{∞}(G_{t})}^{m,l}(u_{t}), where t ∈ G is a parameter. For a single function v, the statement q ∉ WF_{sc}^{m,l}(v) is equivalent to the existence of A ∈ Ψ_{sc}^{m,−l}(X) which is elliptic at q and such that Av ∈ L^{2}(X), while the statement q ∉ WF_{sc,L^{∞}(G_{t})}^{m,l}(u_{t}) is equivalent to the existence of A as above such that ∥Au_{t}∥_{2} is uniformly bounded.

We begin with a Lemma describing the basic structure of positive commutator estimates.

Lemma 4.1. Suppose a ≥ 0 ∈ x^{−l−1}C^{∞}(^{sc}T^{∗}X) satisfies

(4.2)  ^{sc}H_{p}a = −x^{−l−1}b^{2} + x^{−l−1}e,

with a^{1/2}b, b and e ∈ C^{∞}(^{sc}T^{∗}X). Suppose

A ∈ Ψ_{sc}^{−∞,−l−1}(X), B ∈ Ψ_{sc}^{−∞,−l−1/2}(X) and E ∈ Ψ_{sc}^{−∞,−2l−1}(X)

have principal symbols a, a^{1/2}b and ae respectively, with WF′_{sc}(E) ⊂ supp e, and assume that

(4.3)  a ≤ Cb^{2} on {e = 0} for some C > 0.

Then there exists F ∈ Ψ_{sc}^{−∞,−2l}(X) such that for all u ∈ H_{sc}^{∗,l+1/2}(X),

(4.4) ∥Bu∥^{2} + 2t∥Au∥^{2} ≤ |⟨u, Eu⟩| + |⟨u, F u⟩| + 2|⟨u, A^{∗}A(P − (λ + it))u⟩|.

In particular, if G ⊂ (0, 1]_{t}, and u_{t}, t ∈ G, is a (not necessarily bounded) family in H_{sc}^{∗,l+1/2}(X) such that

WF_{sc,L^{∞}(G_{t})}^{∗,l}(u_{t}) ∩ supp a = ∅,

WF_{sc,L^{∞}(G_{t})}^{∗,l+3/2}((P − (λ + it))u_{t}) ∩ supp a = ∅

and WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) ∩ supp e = ∅,

then WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) ⊂ {b = 0}.

The assumptions t > 0, G ⊂ (0, 1] can be replaced by t < 0, G ⊂ [−1, 0) if we arrange that 0 ≤ a ∈ x^{−l−1}ψ_{0}(p)C^{∞}(^{sc}T^{∗}X) satisfies

scH_{p}a = +x^{−l−1}b^{2} + x^{−l−1}e,

with a^{1/2}b, b and ae ∈ C^{∞}(^{sc}T^{∗}X).

Proof. Let A ∈ Ψ_{sc}^{−∞,−l−1}(X) be a quantization of a, i.e. be such that σ_{∂,−l−1}(A) = x^{l+1}a. From the symbol calculus, and (4.2),

(4.5)  ix^{l+1/2}[A^{∗}A, P]x^{l+1/2} = 2x^{l+1/2}(B^{∗}B + E + F)x^{l+1/2},

where

(4.6)

F ∈ Ψ_{sc}^{−∞,−2l}(X), WF′_{sc}(F) ⊂ supp(x^{l+1}a),
B ∈ Ψ_{sc}^{−∞,−l−1/2}(X) and σ_{∂,−l−1/2}(B) = ba^{1/2}.

For v ∈ H_{sc}^{∗,l+1}(X), t ∈ ℝ,

(4.7)  ⟨v, i[A^{∗}A, P]v⟩ = 2 Im⟨v, A^{∗}A(P − (λ + it))v⟩ − 2t∥Av∥^{2}.

Now if u ∈ H_{sc}^{∗,l+1/2}(X), consider this for v = u_{r} = ϕ(x/r)u where ϕ ∈ C^{∞}(ℝ) is identically 1 on [2, +∞) and identically 0 on [0, 1]. Writing (P − (λ + it))u_{r} = ϕ(x/r)(P −(λ+it))u+[P, ϕ(x/r)]u, observe that A^{∗}A[P, ϕ(x/r)] ∈ Ψ_{sc}^{−∞,∞}(X) is, for r ∈ (0, 1], uniformly bounded in Ψ_{sc}^{−∞,−2l−1}(X), and indeed converges to 0 strongly as an operator H_{sc}^{∗,l+1/2}(X) ⟶ H_{sc}^{∗,−l−1/2}(X), while u_{r} → u in H_{sc}^{∗,l+1/2}(X). Taking the limit r → 0, we deduce that

(4.8)  ⟨u, i[A^{∗}A, P]u⟩ = 2 Im⟨u, A^{∗}A(P − (λ + it))u⟩ − 2t∥Au∥^{2}.

Combining this with (4.5), we conclude that (4.4) holds.

Now consider the uniform statement for the family. Suppose first that

WF_{sc,L^{∞}(G_{t})}^{∗,l+2}((P − λ + it)u_{t}) ∩ supp a = ∅.

Since t > 0, the second term on the left in (4.4) can be dropped. Since u_{t} is microlocally uniformly bounded in H_{sc}^{0,l}(X) on supp a and f_{t} = (P − (λ + it))u_{t} is microlocally uniformly bounded in H_{sc}^{0,l+2}(X) by our assumption, it follows that the right hand side of (4.4) stays bounded as t → 0. Thus, Bu_{t} is uniformly bounded in L_{sc}^{2}(X). This proves that supp b is disjoint from WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}).

In the general case, when we only assume that WF_{sc,L^{∞}(G_{t})}^{∗,l+3/2}((P − λ + it)u_{t}) ∩ supp a = ∅, we need to remove u from the last term of (4.4). To do so, we apply Cauchy-Schwarz to the last term and estimate it by

2∥x^{1/2}Au_{t}∥_{2} ∥x^{1/2}A(P − (λ + it))u_{t}∥_{2} ≤

ϵ∥x^{1/2}Au_{t}∥_{2}^{2} + ϵ^{−1}∥x^{−1/2}A(P − (λ + it))u_{t}∥_{2}^{2},

for any ϵ > 0. By the assumption WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) ∩ supp e = ∅, we can find Q ∈

Ψ_{sc}^{0,0}(X) such that WF′_{sc}(Q)∩WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) = ∅ (hence Qu_{t} is uniformly bounded

in H_{sc}^{∗,l+1/2}(X)), while supp e ∩ WF′_{sc}(Id −Q) = ∅, so e = 0 on WF′_{sc}(Id −Q). By assumption (4.3), we have a ≤ Cb^{2} on WF′_{sc}(Id −Q). Then we may estimate

(4.9) ϵ⟨x^{1/2}Au_{t}, x^{1/2}Au_{t}⟩ = ϵ(⟨x^{1/2}AQ^{2}u_{t}, x^{1/2}Au_{t}⟩ + ⟨x^{1/2}A(Id −Q^{2})u_{t}, x^{1/2}Au_{t}⟩)

≤ ϵC′(∥x^{1/2}AQu_{t}∥_{2}^{2} + ∥Bu_{t}∥_{2}^{2} + |⟨F̃u_{t}, u_{t}⟩|),

for some C′ > 0 and F̃ ∈ Ψ_{sc}^{∗,−2l}(X). Choose ϵ = (2C′)^{−1}. Then we may remove the ∥Bu_{t}∥_{2}^{2} term from the right hand side of (4.4) by replacing ∥Bu_{t}∥^{2} by 1/2∥Bu_{t}∥^{2} on the left hand side, obtaining

\begin{matrix}\frac{∥Bu_{t}∥^{2}}{2} + t∥Au_{t}∥^{2} \\ ≤ C′′(∥x^{1/2}AQu_{t}∥_{2}^{2} + |⟨u_{t}, F u_{t}⟩| + ϵ^{−1}∥x^{−1/2}A(P − (λ + it))u_{t}∥^{2}). \end{matrix} ; (4.10)

The argument of the preceeding paragraph now applies and completes the proof. □

Remark 4.2. The conclusions of Lemma 4.1 hold if we weaken the assumption that u ∈ H_{sc}^{∗,l+1/2}(X) to u ∈ H_{sc}^{∗,k}(X), for some k ∈ ℝ, and WF_{sc}^{∗,l+1/2}(u) ∩ supp a = ∅. We may then apply operators C ∈ Ψ_{sc}^{∗,−l−1/2}(X) with essential support contained in supp a to u. This is done by the prescription

Cu = CQu + C(Id −Q)u,

where Q ∈ Ψ_{sc}^{0,0}(X) is such that WF′_{sc}(Q) ∩ WF′_{sc}(C) = ∅, and WF′_{sc}(Id −Q) ∩ WF_{sc}^{∗,l+1/2}(u) = ∅. This is independent of the choice of pseudodifferential operator Q. This observation allows the iterative use of the Lemma to gain regularity, as is done below.

If ν has a fixed sign, then so does the coefficient −2νx of ∂_{x} in the Hamilton vector field of p. This enables us to prove the second part of the conclusion of the lemma above without requiring the existence of a t-dependent family u_{t} satisfying various conditions, i.e. we can work directly with t = 0.

Lemma 4.3. Suppose that a, b, e, etc., are as in Lemma 4.1, and suppose that ν > 0 on supp a and that u ∈ H_{sc}^{∗,l}(X), with

WF_{sc}^{∗,l+3/2}((P − λ)u) ∩ supp a = ∅ and

WF_{sc}^{∗,l+1/2}(u) ∩ supp e = ∅.
Then WF_{sc}^{∗,l+1/2}(u) ⊂ {b = 0}.

The same conclusion holds if a ≥ 0, ν < 0 on supp a and

scH_{p}a = x^{−l−1}ψ_{0}(p)b^{2} + x^{−l−1}ψ_{0}(p)e,

with a^{1/2}b, b and e ∈ C^{∞}(^{sc}T^{∗}X).

Remark 4.4. This theorem will be used to analyze functions whose wavefront set is concentrated near a radial point of ^{sc}H_{p}. Away from the radial points, the sign of ν is irrelevant and Theorem 3.1 tells us how the wavefront set of u propagates.

Proof. For r > 0 set χ_{r} = (1 + r/x)^{−1/2} = (\frac{r+x}{x})^{−1/2}. Let A, B, E and F be as before, and let A_{r} = Aχ_{r}, B_{r} = Bχ_{r} and E = Eχ_{r}. We repeat the arguments of the previous proposition with A_{r} in place of A. Now, ^{sc}H_{p}χ_{r}^{2} = −2ν(r/x)(1+r/x)^{−2} = −c_{r}^{2} is negative in ν > 0 and so on supp a. Thus, for r > 0,

(4.11) ix^{l+1/2}[A_{r}^{∗}A_{r}, P]x^{l+1/2} = x^{l+1/2}(B_{r}^{∗}B_{r} + A^{∗}C_{r}^{∗}C_{r}A + E_{r} + F_{r})x^{l+1/2},

where C_{r} is the operator multiplication by c_{r}, F_{r} ∈ Ψ_{sc}^{−∞,−2l+1}(X) for r > 0, is uniformly bounded in Ψ_{sc}^{−∞,−2l}(X) as r → 0 and WF′_{sc}(F_{r}) ⊂ supp(x^{l+1}a) uniformly as r → 0.

Note that A_{r} ∈ Ψ_{sc}^{−l−1/2}(X) for r > 0, and it is uniformly bounded in Ψ_{sc}^{−l−1}(X) as r → 0. Since u ∈ H_{sc}^{∗,l}(X) by our assumption, the proof of (4.8) (with l replaced by l + 1/2) is applicable. Combining this with (4.11), we deduce that for r > 0

(4.12) ∥B_{r}u∥^{2} + ∥C_{r}Au∥^{2} ≤ |⟨u, F_{r}u⟩| + |⟨u, E_{r}u⟩| + 2|⟨u, A_{r}^{∗}A_{r}(P − λ)u⟩|.

We now drop the term ∥C_{r}Au∥^{2}, and apply Cauchy-Schwarz to the last term as in the previous proof, obtaining

(4.13)  \frac{∥B_{r}u∥^{2}}{2} ≤ C′′(|⟨E_{r}u, u⟩| + |⟨u, F̃_{r}u⟩| + ϵ^{−1}∥x^{−1/2}A_{r}(P − λ)u∥^{2}).

The right hand side of (4.13) remains uniformly bounded as r → 0, so we conclude that Bu ∈ H_{sc}^{0,0}(X), which implies the conclusion of the Lemma. □

For basic spectral and scattering theory, partially microlocal estimates suffice, i.e. one does not need full microlocal results. These partially microlocal results (namely, microlocal in the ν variable, global in y and μ) are closely related to the two-body resolvent estimates of Isozaki and Kitada [11], and the corresponding many-body estimates of Gérard, Isozaki and Skibsted [4].

Proposition 4.5. Suppose that, for t ∈ G ⊂ (0, 1], some l > −1, ν_{0} < 0 and r > −1/2, the function (P − (λ + it))u_{t} = f_{t} satisfies

WF_{sc,L^{∞}(G_{t})}^{∗,l}(u_{t}) ∩ {ν < ν_{0}} = ∅ and
WF_{sc,L^{∞}(G_{t})}^{∗,r+1}(f_{t}) ∩ {ν < ν_{0}} = ∅,

then WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) ∩ {ν < ν_{0}} = ∅.

Proof. The key point is that ^{sc}H_{p} is radial at RP(λ) = RP_{+}(λ) ∪ RP_{−}(λ). That is, the tangential component W vanishes there, so we need to exploit the normal component, which is −2νx∂_{x}. We obtain a positive commutator by using a weight x^{−l−1}, since −l − 1 < 0. Fix ϵ > 0, δ > 0, M > 0, and let

(4.14)  a = x^{−l−1}χ(ν)ψ̃(x)ψ_{0}(p) ≥ 0

where ψ̃ ∈ C_{c}^{∞}(ℝ) is identically 1 near 0 and is supported in a bigger neighbourhood of 0 (it is simply a cutoff near ∂X), ψ_{0} ∈ C_{c}^{∞}(ℝ; [0, 1]) supported in (λ − δ, λ + δ), χ ∈ C_{c}^{∞}(ℝ; [0, 1]) is identically one on (−∞, ν_{0}−3ϵ), vanishes on (ν_{0}−ϵ, ∞), χ′ ≤ 0, and χ(t) = e^{M/(t−(ν_{0}−ϵ))} on (ν_{0} − 2ϵ, ν_{0} − ϵ). Then

\begin{aligned}& ^{sc}H_{p}a = 2((l + 1)νχ(ν) + |μ|_{y}^{2}χ′(ν))x^{−l−1}ψ_{0}(p) = −b^{2}x^{−l−1}, \\ (4.15) \end{aligned}

and b is C^{∞} by the construction of χ. Thus, if r < l + 3/2, then Lemma 4.1 gives us the result (with ν_{0} replaced by ν_{0} − ϵ). Otherwise, from the Lemma we gain a power of x^{1/2} in the wavefront set estimate for u_{t}, i.e. , we have

WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) ∩ {ν < ν_{0} − ϵ} = ∅.

We may now repeat the argument with l replaced by l+1/2. Iterating the argument a finite number of times (cf. the remark after the proof of Lemma 4.1), then sending ϵ → 0, we obtain our estimate. □

A similar argument works in ν ≥ 0, but this time we only need to assume boundedness in H_{sc}^{∗,l}(X) for an arbitrary l ∈ ℝ in this region, i.e. we do not need l ∈ (−1, −1/2). On the other hand, we are only able to deduce boundedness in H_{sc}^{∗,r}(X) for r < −1/2. In addition, we need to assume that the desired boundedness, i.e. in H_{sc}^{∗,r}(X) for r < −1/2, holds for ν near ν_{0} > 0.

Proposition 4.6. Suppose f_{t} = (P − (λ + it))u_{t}, where t ∈ G ⊂ (0, 1], ν_{0} > 0, l ∈ ℝ and r < −1/2. If we have

WF_{sc,L^{∞}(G_{t})}^{∗,l}(u_{t}) ∩ {ν ≥ ν_{0}} = ∅,

WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) ∩ {ν = ν_{0}} = ∅ and
WF_{sc,L^{∞}(G_{t})}^{∗,r+1}(f_{t}) ∩ {ν ≥ ν_{0}} = ∅,
then WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) ∩ {ν ≥ ν_{0}} = ∅.

Proof. The proof is nearly identical to that of Proposition 4.5. We may assume l < −1 and l < r without loss of generality. Also, the estimate is clear away from Σ(λ) by elliptic regularity, so we may restrict our attention to a small neighbourhood O of Σ(λ) whose closure \overline{O} is compact. Take a as in (4.14), where now χ′ ≥ 0, χ vanishes on (−∞, ν_{0} − ϵ], is identically 1 on [ν_{0} + ϵ, +∞), and given by e^{M/(ν−(ν_{0}+ϵ))} on (ν_{0}, ν_{0}+ϵ). We assume that ϵ > 0 is so small that WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t})∩\{χ′ ≠ 0\}∩\overline{O} = ∅ and WF_{sc,L^{∞}(G_{t})}^{∗,r+1}(f_{t}) ∩ \{χ ≠ 0\} ∩ \overline{O} = ∅. Such a choice is possible since the wave front set is closed and \overline{O} is compact. Now l+1 < 0 and ν > 0, so the sign of the first term in (4.15) is unchanged, while the sign of the second term is reversed. Hence, the second term becomes an error term, e, which satisfies (4.3). By construction we have WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) ∩ supp e = ∅. Thus, we can apply Lemma 4.1, following the argument of the previous Proposition, to complete the proof. □

Versions of these results with no parameter t can also be obtained. In this case, there is no distinction between ν > 0 and ν < 0.

Proposition 4.7. Let ν_{0} < 0 and l > −1/2. Suppose that P u = f, where

WF_{sc}^{∗,l}(u) ∩ {ν ≤ ν_{0}} = ∅ and WF_{sc}^{∗,r+1}(f) ∩ {ν ≤ ν_{0}} = ∅.

Then

WF_{sc}^{∗,r}(u) ∩ {ν ≤ ν_{0}} = ∅.

A similar result holds with ν_{0} > 0 with all inequalities ν ≤ ν_{0} replaced by ν ≥ ν_{0}.

Proof. The proof uses the symbol a from Proposition 4.5, and proceeds via a regularization argument similar to Lemma 4.3. Now, however, ^{sc}H_{p}a and ^{sc}H_{p}χ_{r}^{2} have opposite signs, so the two terms cannot be treated separately. Instead, we use the argument of the lemma with a replaced by x^{1/2}a and χ_{r} = (1 + r/x)^{−1/2} replaced by x^{−1/2}χ_{r} = (x + r)^{−1/2}. Now ^{sc}H_{p}(x^{1/2}a) and ^{sc}H_{p}(xχ_{r}^{2}) have the same sign, so the proof of Lemma 4.3 is applicable, and completes the proof of the proposition. Note that the power of the factor of x in x^{1/2}a is x^{−l−1/2}; this is where we use that l > −1/2. □

The combination of Theorem 3.1 and Proposition 4.7 leads to

Proposition 4.8. Suppose that l > −1/2, λ_{0} ∉ Cv(V). Then there exist operators F_{j} ∈ Ψ_{sc}^{−1,1}(X), j = 1, …, N, and constants C > 0, δ > 0, such that for u ∈ H_{sc}^{m,l}(X), (P − λ)u ∈ H_{sc}^{m,l+1}(X), λ ∈ (λ_{0} − δ, λ_{0} + δ),

∥u∥_{H_{sc}^{m,l}(X)} ≤ ∑_{j=1}^{N} ∥F_{j}u∥_{H_{sc}^{m,l}(X)} + C∥(P − λ)u∥_{H_{sc}^{m,l+1}(X)}.

Proof. Choose a small ν_{0} > 0 so that the Hamiltonian flow never vanishes for |ν| ≤ 2ν_{0}, on the chacteristic surfaces Σ(λ), λ ∈ (λ_{0}−δ, λ_{0}+δ). We then decompose the identity operator in the form Id = Q_{0} + Q_{1} + B_{+} + B_{−}, where all operators are in Ψ_{sc}^{0,0}(X), and are microlocalized as follows: Q_{0} is microsupported in |ν| ≤ 2ν_{0}, B_{±} are of the form that arise in the proof of Proposition 4.7, thus microsupported in {±ν > ν_{0}}, and Q_{1} is microsupported away from the characteristic sets Σ(λ). Since the Hamilton flow does not vanish on the microsupport of Q_{0}, we may write Q_{0} as a commutator with P − λ, modulo lower order terms, and hence obtain an estimate

∥Qu∥_{H_{sc}^{m,l}(X)} ≤ C(∥(P − λ)u∥_{H_{sc}^{m,l+1}(X)} + |⟨Eu, u⟩|),

where E ∈ Ψ_{sc}^{2m,−2l}(X) satisfies WF′_{sc}(E) ⊂ {|ν| ≥ ν_{0}}. We add to this a sufficiently large multiple of the estimates for ∥B_{±}u∥_{H_{sc}^{m,l}(X)}, obtained from the proof of Proposition 4.7, and the estimate for ∥Q_{1}u∥_{H_{sc}^{m,l}(X)} obtained from ellipticity. This allows one to absorb the error term |⟨Eu, u⟩| above. □

This immediately yields the usual results on the spectrum of P. For the last statement we also need a unique continuation result.

Proposition 4.9. (See Theorem 3.2) For λ ∉ Cv(V) the space W of L^{2} eigenfunctions with eigenvalue λ is a finite dimensional subspace of Ċ^{∞}(X). The pure point spectrum, σ_{pp}(P), is disjoint from (sup Cv(V), +∞) and can only accumulate at Cv(V).

Proof. Let F_{k} be as in Proposition 4.8. Since these operators are compact, there exists a finite dimensional subspace W_{0} of W such that for u ∈ W orthogonal to W_{0}, ∥F_{k}u∥_{H_{sc}^{0,0}(X)} ≤ ∥u∥_{H_{sc}^{0,0}(X)}/2N. Suppose that u ∈ W, ∥u∥_{H_{sc}^{0,0}(X)} = 1. Applying the proposition with λ_{0} = λ we deduce that

1 = ∥u∥_{H_{sc}^{0,0}(X)} ≤ ∑_{k=1}^{N} ∥F_{k}u∥_{H_{sc}^{0,0}(X)} ≤ 1/2,

which is a contradiction. Hence W = W_{0}, so W is finite dimensional.

More generally, let W̃ be the sum of the L^{2} eigenspaces of P with eigenvalue λ̃ ∈ [λ_{0} − δ/2, λ_{0} + δ/2]. By compactness of the F_{k}, there is a finite dimensional subspace W̃_{0} of W̃ such that for u ∈ W̃ orthogonal to W̃_{0}, ∥F_{k}u∥_{H_{sc}^{0,0}(X)} ≤ ∥u∥_{H_{sc}^{0,0}(X)}/2N. Again applying Proposition 4.8 with λ̃ in place of λ in the last term, ∥(P − λ̃)u∥_{H_{sc}^{m,l+1}(X)} = 0, shows that W̃ = W̃_{0}, so W̃ is also finite dimensional.

That an L^{2} eigenfunction u is in Ċ^{∞}(X) follows from the following Proposition. The statement that σ_{pp}(P) is disjoint from (sup Cv(V), +∞) follows from the unique continuation theorem of Froese and Herbst [3], as generalized to scattering metrics in [17]. □

Proposition 4.10. (Theorem 3.2) Suppose that u ∈ C^{−∞}(X), (P − λ)u = 0, WF_{sc}^{m,l}(u) ∩ RP_{−}(λ) = ∅ for some l > −1/2. Then u ∈ Ċ^{∞}(X).

Proof. This is essentially Isozaki’s uniqueness theorem from [10], Lemma 4.5, proved the same way in this setting. From Theorem 3.1 and Proposition 4.7, we have

(4.16) WF_{sc}(u) ⊂ {ν ≥ a(λ) > 0} and u ∈ H_{sc}^{m′,l′}(X) ∀ m′ ∈ ℝ, l′ < −1/2,

where a(λ) is as in Remark 1.11. Let l ∈ (−1/2, 0) and let ϕ ∈ C^{∞}(ℝ) be 0 on (−∞, 1] and 1 on [2, ∞). For r > 0 let

(4.17)  χ_{r}(x) = r^{−2l−1} ∫_{0}^{x/r} ϕ^{2}(s)s^{−2l−2} ds.

Thus, χ_{r} ∈ C_{c}^{∞}(int(X)) and

(4.18)  x^{2}∂_{x}χ_{r}(x) = x^{−2l}ϕ^{2}(x/r).

Note that χ_{r} is not uniformly bounded in Ψ_{sc}^{m′,l′}(X) for any m′ and l′, but x^{2}∂_{x}χ_{r} is. Since χ_{r} enters the commutator [χ_{r}(x), P] only via x^{2}∂_{x}χ_{r}, that boundedness will suffice for us. Now, by [13, Equation 3.7]

(4.19)  Δ = (x^{2}D_{x})^{2} + i(n − 1)x^{3}D_{x} + x^{2}Δ_{h} + x^{3} Diff_{b}^{2}(X),

so

(4.20)  −i[χ_{r}(x), P] = 2x^{−2l}ϕ^{2}(x/r)(x^{2}D_{x}) + F′_{r}

where F′_{r} is bounded in Ψ_{scc}^{1,−2l+1}(X). Let ψ ∈ C_{c}^{∞}(ℝ) be supported close to λ and be identically 1 near λ. Let ρ ∈ C^{∞}(ℝ) be 0 on (−∞, a(λ)/4) and be 1 on (a(λ)/2, ∞). Let B ∈ Ψ_{sc}^{−∞,0}(X) and E ∈ Ψ_{sc}^{−∞,0}(X) have symbols σ_{∂,0}(B) = \sqrt{ν}ρ(ν)ψ(p) ∈ C^{∞} and σ_{∂,0}(E) = ν(1 − ρ^{2}(ν))ψ^{2}(p). Then

(4.21)  −i[χ_{r}(x)ψ^{2}(P), P] = 2x^{−l}ϕ(x/r)(B^{2} + E)ϕ(x/r)x^{−l} + F_{r}

where WF′_{sc}(E) is disjoint from {ν ≥ a(λ)/2} and F_{r} is bounded in Ψ_{sc}^{−∞,−2l+1}(X). Now, for r > 0

(4.22)  ⟨u, [χ_{r}(x), P]u⟩ = −2i Im⟨u, χ_{r}(x)(P − λ)u⟩ = 0.

Hence,

(4.23)  ∥x^{−l}ϕ(x/r)Bu∥^{2} ≤ |⟨x^{−l}ϕ(x/r)u, Ex^{−l}ϕ(x/r)u⟩| + |⟨u, F_{r}u⟩|.

In view of (4.16) the right hand side stays bounded as r → 0, so we conclude that x^{−l}Bu ∈ L_{sc}^{2}(X). Then, by (4.16), it follows that u ∈ H_{sc}^{∞,l}(X). Since l > −1/2, we conclude from Theorem 3.1 and Proposition 4.7 that u ∈ Ċ^{∞}(X). □

As mentioned above, χ_{r}(x) is not bounded in Ψ_{sc}^{m′,l′}(X) for any m′ and l′, so the place where we have used the assumption (P − λ)u = 0 is really in the elimination of the term on the right hand side of (4.22) from the right hand side of (4.23).

Theorem 4.11. (Theorem 3.3) The resolvent R(λ + it), t > 0, λ ∉ Λ = Cv(V) ∪ σ_{pp}(P) extends continuously to the real axis, i.e. R(λ + i0) exists, as a bounded operator H_{sc}^{m,r}(X) ⟶ H_{sc}^{m+2,l}(X) for any r > 1/2 and l < −1/2.

Proof. Let f ∈ H_{sc}^{m,r}(X), for r > 1/2, let −1 < l < −1/2, and suppose that u_{t} = R(λ + it)f is not bounded in H_{sc}^{m+2,l}(X) as t → 0. We can then take a sequence t_{j} → 0 with ∥u_{t_{j}}∥_{H_{sc}^{m+2,l}(X)} → ∞. Now consider v_{t_{j}} = u_{t_{j}}/∥u_{t_{j}}∥_{H_{sc}^{m+2,l}(X)}, so v_{t_{j}} remains bounded in H_{sc}^{m+2,l}(X), and (P − (λ + it))v_{t_{j}} → 0 in H_{sc}^{m,r}(X). We may then pass to a convergent subsequence in H_{sc}^{m+2−δ,l−δ}(X), δ > 0, with limit v. Thus (P − λ)v = 0, and by Proposition 4.5, RP_{−}(λ) ∩ WF_{sc,L^{∞}(G_{t})}^{∗,l′}(v_{t}) = ∅ for some l′ > −1/2. By the t-dependent part of Theorem 3.1 on the propagation of singularities WF_{sc,L^{∞}(G_{t})}^{∗,l′}(v_{t}) ⊂ {ν > 0}, hence the same holds for v. The uniqueness result, Theorem 3.2 then shows that v = 0. However this contradicts the fact that ∥v_{t_{j}}∥_{H_{sc}^{m+2,l′} (X)} = 1. Hence, there exists C > 0 such that ∥u_{t}∥_{H_{sc}^{m+2,l}(X)} ≤ C for all t. Then, by Proposition 4.5 and the t-dependent part of

Theorem 3.1, WF_{sc,L^{∞}(G_{t})}^{∗,l′}(u_{t}) ⊂ {ν > 0}.

Now suppose that t_{j} → 0. By decreasing m + 2 and l, we can pass to a convergent subsequence with limit u, which then satisfies (P − λ)u = f, and WF_{sc}^{∗,l′}(u) ⊂ {ν > 0}. Taking any other sequence t′_{j} and any convergent subsequence, we obtain another distribution u′ with the same properties, hence by considering their difference and using Theorem 3.2, u = u′. If now t′′_{j} is a sequence such that t′′_{j} → 0 and ∥u_{t′′_{j}} − u∥_{H_{sc}^{m+2,l}(X)} ≥ ϵ > 0, then taking a subsequence of t′′_{j} with u_{t′′_{j}} converging in H_{sc}^{m+2−δ,l−δ}(X) to some u′′, we deduce that u = u′′, which is a contradiction. This shows convergence of R(λ + it)f as t → 0, and similar arguments easily give boundedness as an operator, and continuity in λ as well. □

We next refine this theorem by combining it with the propagation estimates. To do so we need a microlocal version of Proposition 4.5, which we state afterwards.

Theorem 4.12. (Theorem 3.4) If λ ∉ Λ and f ∈ Ċ^{∞}(X) then WF_{sc}(R(λ + i0)f) ⊂ Φ_{+}(RP_{+}(λ)). Moreover R(λ + i0) extends by continuity to v ∈ C^{−∞}(X) with WF_{sc}(v) ∩ Φ_{−}(RP_{−}(λ)) = ∅, and for such v, R(λ + i0)v satisfies (3.7).

Proof. Let u = R(λ+i0)f. It follows from Proposition 4.5 that WF_{sc}(u) ⊂ {ν ≥ 0}. Then Theorem 3.1 shows that WF_{sc}(u) ⊂ Φ_{+}(RP_{+}(λ)). Indeed, given q ∈ Σ(λ), let γ be the bicharacteristic through q. Then q_{0} = lim_{t→−∞} γ(t) ∈ RP_{+}(λ) ∪ RP_{−}(λ). If q_{0} ∈ RP_{+}(λ), there is nothing to prove, since then q ∈ Φ_{+}(RP_{+}(λ)). If q_{0} ∈ RP_{−}(λ), there exist points on γ where ν is negative, hence these are not in WF_{sc}(u). But then by Theorem 3.1, q ∉ WF_{sc}(u).

A duality argument immediately gives the extension of R(λ + i0) to the class of distributions described above. The wave front set bound is a consequence of Theorem 3.1 and the following fully microlocal family version of Proposition 4.5, stated below. □

Proposition 4.13. Suppose that q ∈ RP_{−}(λ), f_{t} = (P − (λ + it))u_{t}, for t ∈ G ⊂ (0, 1], and O is a W -balanced neighbourhood (see Definition 1.7) of q then for r > −1/2, and l > −1,

O ∩ WF_{sc,L^{∞}(G_{t})}^{∗,r+1}(f_{t}) = ∅, O ∩ WF_{sc,L^{∞}(G_{t})}^{∗,l}(u_{t}) = ∅ and

(O \ Φ_{+}({q})) ∩ WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) = ∅

implies O ∩ WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) = ∅.

Proof. Let q = (y_{0}, ν_{0}, 0), ν_{0} < 0, in our local coordinates. Due to our assumptions, and Theorem 3.1, we only need to make a commutator positive on Φ_{+}({q}).

For l > −1 consider a symbol a of the form

(4.24)  a = x^{−l−1}χ(ν)ϕ(y, μ)ψ̃(x)ψ_{0}^{2}(p) ≥ 0

where ψ̃ ∈ C_{c}^{∞}(ℝ) is identically 1 near 0 and is supported in a bigger neighbourhood of 0 (it is simply a cutoff near ∂X), ψ_{0} ∈ C_{c}^{∞}(ℝ; [0, 1]) supported in (λ − δ, λ + δ), ϕ ≥ 0 is C^{∞}, supported near (y_{0}, 0), identically 1 in a neighbourhood of this point, χ ∈ C_{c}^{∞}(ℝ; [0, 1]) supported in [ν_{0} − ϵ, ν_{0} + ϵ], identically 1 near ν_{0}, χ′ ≤ 0 on [ν_{0} − ϵ/2, ν_{0} + ϵ], χ′ ≥ 0 on [ν_{0} − ϵ, ν_{0} − ϵ/2] and χ(ν) = e^{M/(ν−(ν_{0}+ϵ))} on (ν_{0} + ϵ/2, ν_{0} + ϵ). In addition, we choose ϵ > 0 so that [ν_{0} − 2ϵ, ν_{0} + 2ϵ] × supp dϕ ⊂ O \ Φ_{+}({q}). This can be achieved by fixing ϕ and making ϵ > 0 small since \{ν_{0}\} × supp dϕ is disjoint from the compact set Φ_{+}(\{q\}) ∩ \overline{O}.

Let ρ ∈ C_{c}^{∞}(ℝ) be identically 0 on (−∞, ν_{0} − 3ϵ/8), identically 1 on (ν_{0} − ϵ/4, +∞). Notice that χ′ ≤ 0 on the support of ρ. Then, near x = 0,

\begin{matrix}^{sc}H_{p}a = 2((l + 1)νχ(ν) + |μ|_{y}^{2}χ′(ν))ϕ(y, μ)x^{−l−1}ψ_{0}^{2}(p) + x^{−l−1}χ(ν)ψ_{0}^{2}(p) ^{sc}H_{p}ϕ \\ = −b^{2}x^{−l−1} + ex^{−l−1}, \\ b^{2} = ( − 2ν(l + 1)χ(ν) − 2ρ^{2}(ν)|μ|_{y}^{2}χ′(ν))ϕ(y, μ)ψ_{0}^{2}(p), \\ e = (2(1 − ρ^{2}(ν))|μ|_{y}^{2}χ′(ν))ϕ(y, μ) + χ(ν) ^{sc}H_{p}ϕ)ψ_{0}^{2}(p), \end{matrix} (4.25)

and b is C^{∞} by the construction of χ. Note that supp e is disjoint from WF_{sc}^{∗,r}(u), and a, b, e satisfy (4.3). Thus, by Lemma 4.1, we gain half a power of x in our regularity of u_{t} near q, i.e. , we get WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) = ∅ near q. By The-

orem 3.1 this gives O ∩ WF_{sc,L^{∞}(G_{t})}^{∗,l+1/2}(u_{t}) = ∅. Iterating the argument, we get O ∩ WF_{sc,L^{∞}(G_{t})}^{∗,r}(u_{t}) = ∅ after a finite number of steps. □

5. Microlocal eigenfunctions

With each zero q of the vector field W, when λ > V (π(q)), we associate spaces of microlocally incoming, respectively outgoing eigenfunctions. These are microfunctions near q with respect to the scattering wavefront set, satisfy the eigenfunction equation microlocally near q, and have wave front set in ν < ν(q), resp. ν > ν(q).

For a W -balanced neighbourhood O ∋ q, as in Definition 1.7, the spaces of incoming and outgoing microlocal eigenfunctions, Ẽ_{mic,±}(q, λ), are defined in (I.15). The propagation of singularities in regions of real principal type (i.e. , where W ≠ 0) shows that stronger restrictions on WF_{sc}(u) follow directly.

Lemma 5.1. If O ∋ q is, for q ∈ RP(λ), a W -balanced neighbourhood then for every u ∈ Ẽ_{mic,±}(O, λ)

WF_{sc}(u) ∩ O ⊂ Φ_{±}({q}) and
WF_{sc}(u) ∩ O = ∅ ⟺ q ∉ WF_{sc}(u).

Thus, we could have defined Ẽ_{mic,±}(O, λ) by strengthening the restriction on the wavefront set to WF_{sc}(u)∩O ⊂ Φ_{±}({q}). With such a definition there is no need for O to be W -balanced; the only relevant bicharacteristics would be those contained in Φ_{±}({q}).

Proof. We may assume that q ∈ RP_{+}(λ). For the sake of definiteness also assume that u ∈ Ẽ_{mic,+}(O, λ); the other case follows similarly. Consider ζ ∈ O \ {q}. If ν(ζ) < ν(q), then ζ ∉ WF_{sc}(u) by the definition of Ẽ_{mic,+}(O, λ), so we may suppose that ν(ζ) ≥ ν(q). Since q ∈ Φ_{+}({q}) we may also suppose that ζ ≠ q.

Let γ : ℝ → Σ(λ) be the bicharacteristic through ζ with γ(0) = ζ. As O is W -convex, and WF_{sc}((P − λ)u) ∩ O = ∅, Theorem 3.1 shows that

ζ ∈ WF_{sc}(u) ⇒ γ(ℝ) ∩ O ⊂ WF_{sc}(u).

As O is W -balanced, there exists ζ′ ∈ \overline{γ(ℝ)} ∩ O such that ν(ζ′) = ν(q).

If ζ′ = γ(t_{0}) for some t_{0} ∈ ℝ, then for t < t_{0}, ν(γ(t)) < ν(γ(t_{0})) = ν(q), and for sufficiently small |t − t_{0}|, γ(t) ∈ O as O is open. Thus, γ(t) ∉ WF_{sc}(u) by the definition of Ẽ_{mic,+}(O, λ), and hence we deduce that ζ ∉ WF_{sc}(u).

On the other hand, if ζ′ = lim_{t→−∞} γ(t), then ζ′ ∈ RP(λ), and ζ′ ∈ O, so the fact that O is W -balanced shows that ζ′ = q, hence ζ ∈ Φ_{+}({q}).

The last statement of the lemma follows from Theorem 3.1, and the fact that WF_{sc}(u) is closed. □

Corollary 5.2. Let O be a W -balanced neighbourhood of q ∈ RP_{+}(λ). If π(q) is a minimum of V |_{Y} and u ∈ Ẽ_{mic,+}(O, λ) then WF_{sc}(u) ∩ O ⊂ {q}. If π(q) is a maximum of V |_{Y} and u ∈ Ẽ_{mic,±}(O, λ) then WF_{sc}(u) ∩ O ⊂ L_{±}. Here L_{+} is that one of the Legendrians L_{1}, L_{2} from Proposition 1.5 which satsifies L_{+} ⊂ {ν ≥ ν(q)}, and L_{−} is the other one of these.

To avoid truly microlocal arguments in the analysis of these spaces, such as microlocal solvability along the lines of Hörmander [8], we introduce, as a technical device, an operator P̃ which arises from P by altering V appropriately, and use global analysis of P̃.

Lemma 5.3. Fix λ > min V_{0}, and ν̃ > 0, and set K = V_{0}^{−1}((−∞, λ − ν̃^{2}]). There exists Ṽ ∈ C^{∞}(X) with Ṽ_{0} Morse such that the minima of V_{0} and Ṽ_{0} coincide, V and Ṽ themselves coincide in a neighbourhood of their minima, Ṽ_{0} ≥ V_{0}, Ṽ |_{K} = V |_{K} and no maximum value of Ṽ lies in the interval (λ − ν̃^{2}, λ).

Let P̃ = Δ+Ṽ be the corresponding Hamiltonian, Σ̃(λ) its characteristic variety at eigenvalue λ and \begin{matrix} \\ RP & + \end{matrix}(λ) the set of its radial points in ν > 0. If q ∈ \begin{matrix} \\ RP & + \end{matrix}(λ) with ν(q) < ν̃, then π(q) is a minimum of Ṽ and the full symbols of P and P̃ coincide in a neighbourhood of q; in particular,

(5.1)  Σ(λ) ∩ {ν ≥ ν̃} = Σ̃(λ) ∩ {ν ≥ ν̃}.

Proof. It suffices to modify V in small neighbourhood U_{z} of each maximum z of V_{0} with value in the range (λ − ν̃^{2}, λ] (if there are any). We do this so that the new potential has Ṽ > λ − ν̃^{2} on these sets, has only one critical point in each and at that maximum takes a value greater than λ.

Now q ∈ \begin{matrix} \\ RP & + \end{matrix}(λ) implies that z = π(q) is a critical point of Ṽ_{0} and ν(q)^{2}+Ṽ (z) = λ. Suppose that z is a maximum of Ṽ and 0 < ν(q) < ν̃. Then ν(q)^{2} + Ṽ (z) = λ shows that λ − ν̃^{2} < Ṽ (z) < λ, which is contradicted by the construction of Ṽ . Thus, z is a minimum of Ṽ .

Next, on Σ(λ) ∩ {ν ≥ ν̃}, ν^{2} + |μ|_{y}^{2} + V_{0} = λ, hence V_{0} ≤ λ − ν̃^{2}, so V_{0} = Ṽ_{0}, and therefore Σ(λ) ∩ {ν ≥ ν̃} ⊂ Σ̃(λ). With the converse direction proved similarly, (5.1) follows. □

Remark 5.4. The point of this Lemma is that it allows one to assume, in any argument concerning q ∈ RP_{+}(λ), that there is no q′ ∈ Max_{+}(λ) with ν(q′) < ν(q). The virtue of this is illustrated in the proof of the following continuation result.

Lemma 5.5. Suppose u ∈ C^{−∞}(X) satisfies

WF_{sc}(u) ⊂ {ν ≥ ν_{1}} and WF_{sc}((P − λ)u) ⊂ {ν ≥ ν_{2}},

for some 0 < ν_{1} < ν_{2}, then there exists ũ ∈ C^{−∞}(X) with WF_{sc}(u − ũ) ⊂ {ν ≥ ν_{2}} and (P − λ)ũ ∈ Ċ^{∞}(X).

Proof. One is tempted to try to solve this by adding R(λ + i0)((P − λ)u) to u. This does not quite work, however, since the wavefront set of this is contained in WF_{sc}((P − λ)u) ∪ Φ_{+}(WF_{sc}((P − λ)u ∩ Σ(λ))) ∪ Φ_{+}(RP_{+}(λ)), and only the first two sets are contained in {ν ≥ ν_{2}}. The third is contained in the set Min_{+}(λ), together with the flowouts along L_{2} from Max_{+}(λ). The flowouts are a nuisance here because they ‘bump into’ the rest of the wavefront set at {ν ≥ ν_{2}}. To avoid dealing with this problem, we use the operator P̃ where such points are eliminated.

Let ν̃ be such that ν_{1} < ν̃ < ν_{2}, and ν̃ is sufficiently close to ν_{1} so that there are no critical points in Σ(λ) with ν ∈ [ν̃, ν_{2}). Choose an operator A ∈ Ψ_{sc}^{0}(X) with WF′_{sc}(Id −A) ∩ Σ(λ) ⊂ {ν ≥ ν̃} and WF_{sc}(A) ⊂ {ν ≤ ν_{2}}. Then f = (P − λ)Au has wavefront set confined to Σ(λ) ∩ {ν̃ < ν < ν_{2}}.

Let Ṽ be as in Lemma 5.3, and let Π̃ be the orthogonal projection off the L^{2}- nullspace of P̃ − λ; Id −Π̃ is a finite rank projection onto a subspace of Ċ^{∞}(X). By (5.1), WF_{sc}(f) ⊂ Σ̃(λ) ∩ {ν̃ < ν < ν_{2}}. Now consider v = R̃(λ + i0)Π̃f where R̃(λ) is the resolvent for P̃. This has wavefront set in Σ̃(λ) ∩ {ν > 0}. Furthermore in ν < ν̃, where f has no wavefront set, its only wavefront set is associated to critical points. These are all minima, by construction, so in ν < ν̃, WF_{sc}(v) only has these as isolated points. Using microlocal cut-offs near these points we may excise the wavefront set and so obtain a solution of (P̃ − λ)v′ = f + f′, where f′ ∈ Ċ^{∞}(X) and WF_{sc}(v′) is confined to ν ≥ ν̃ so

WF_{sc}(v′) ⊂ Σ̃(λ) ∩ {ν ≥ ν̃} = Σ(λ) ∩ {ν ≥ ν̃}.

Since

WF_{sc}((P̃ − P)v′) ⊂ Σ(λ) ∩ {ν ≥ ν̃} ∩ WF′_{sc}(P̃ − P) = ∅,

it follows that (P −λ)v′ = f +f′′ with f′′ ∈ Ċ^{∞}(X). Finally then set ũ = Au−v′. By construction this reduces to u microlocally in ν < ν̃ and satisfies (P −λ)ũ ∈ Ċ^{∞}(X). Since there are no critical points with ν ∈ [ν̃, ν_{2}) it follows from Theorem 3.1 that WF_{sc}(u − ũ) is contained in {ν ≥ ν_{2}}. □

If O_{1} and O_{2} are two W -balanced neighbourhoods of q then

(5.2)  O_{1} ⊂ O_{2} ⟹ Ẽ_{mic,±}(O_{2}, λ) ⊂ Ẽ_{mic,±}(O_{1}, λ).

Since {u ∈ C^{−∞}(X); WF_{sc}(u) ∩ O = ∅} ⊂ Ẽ_{mic,±}(O, λ) for all O and this linear space decreases with O, the inclusions (5.2) induce similar maps on the quotients

(5.3) E_{mic,±}(q, λ) = Ẽ_{mic,±}(O, λ)/{u ∈ C^{−∞}(X); WF_{sc}(u) ∩ O = ∅}, O_{1} ⊂ O_{2} ⟹ E_{mic,±}(O_{2}, λ) ⟶ E_{mic,±}(O_{1}, λ).

Lemma 5.6. Provided O_{i}, for i = 1, 2, are W -balanced neighbourhoods of q and the closure of O_{2} contains no radial points other than q, the map in (5.3) is an isomorphism.

Proof. We may assume q ∈ RP_{+}(λ). We prove the lemma first for E_{mic,+}(O_{i}, λ); we need a different method for E_{mic,−}(O_{i}, λ).

If q ∈ Min_{±}(λ), the lemma follows directly from Corollary 5.2, so we may assume that q ∈ Max_{±}(λ). Let L_{+} be that one of the Legendrians L_{1}, L_{2} from Proposition 1.5 with L_{+} ⊂ {ν ≥ ν(q)}.

The map in (5.3) is injective since any element u of its kernel has a representative ũ ∈ Ẽ_{mic,+}(O_{2}, λ) which satisfies q ∉ WF_{sc}(ũ), hence WF_{sc}(ũ) ∩ O_{2} = ∅ by the second half of Lemma 5.1, so u = 0 in E_{mic,+}(O_{2}, λ).

The surjectivity follows from Lemma 5.5. The assumption that \overline{O_{2}} contains no radial points other than q allows us to find an open set O ⊃ \overline{O_{2}} which is W -balanced. Now let A ∈ Ψ_{sc}^{−∞,0}(X) be microlocally the identity on L_{+} ∩ \overline{O_{1}} and supported in a small neighbourhood of L_{+} inside O. Then there exists ν_{2} > ν(q) such that ν > ν_{2} on L_{+} \ O_{1}, and WF′_{sc}(A) \ O_{1} ⊂ {ν ≥ ν_{2}}. Since WF_{sc}(u) ∩ O_{1} ⊂ L_{+}, WF_{sc}(Au − u) ∩ O_{1} = ∅. In addition, WF_{sc}(Au) ⊂ WF′_{sc}(A) ∩ WF_{sc}(u), hence ν ≥ ν(q) on WF_{sc}(Au). Moreover, WF_{sc}(Au − u) ∩ O_{1} = ∅ implies that

WF_{sc}((P − λ)Au) ∩ O_{1} = WF_{sc}((P − λ)Au − (P − λ)u) ∩ O_{1} = ∅,

so WF_{sc}((P − λ)Au) ⊂ WF′_{sc}(A) \ O_{1}, hence is contained in {ν ≥ ν_{2}}. Hence by Lemma 5.5, there exists ũ ∈ C^{−∞}(X) such that ν ≥ ν_{2} on WF_{sc}(ũ − Au) and (P − λ)ũ ∈ Ċ^{∞}(X). In particular, ν ≥ ν(q) in WF_{sc}(ũ), so ũ ∈ Ẽ_{mic,+}(O_{2}, λ). Moreover, ν ≥ ν_{2} on WF_{sc}(ũ − u) ∩ O_{1}, hence by the second half of Lemma 5.1 WF_{sc}(ũ−u)∩O_{1} = ∅, so ũ and u have the same image in E_{mic,+}(O_{1}, λ). This shows surjectivity.

Rather applying Lemma 5.5, we could have used Hörmander’s existence theorem in the real principal type region [8], to find a microlocal solution v of (P − λ)v = (P −λ)Au on O, with WF_{sc}(v) ⊂ Φ_{+}(WF_{sc}((P −λ)Au)), and then taken ũ = Au−v. This method also allows us to deal with E_{mic,−}(O_{i}, λ). □

It follows from this Lemma that the quotient space E_{mic,±}(q, λ) in (5.3) is welldefined, as the notation already indicates, and each element is determined by the behaviour microlocally ‘at’ q.

Corollary 5.7. If q ∈ Min_{+}(λ) then (5.4)

E_{mic,+}(q, λ) ≃ {u ∈ C^{−∞}(X); (P − λ)u ∈ Ċ^{∞}(X), WF_{sc}(u) ⊂ {q}} /Ċ^{∞}(X).

If q ∈ Max_{+}(λ) then every element of E_{mic,+}(q, λ) has a representative u such that (P − λ)u ∈ Ċ^{∞}(X), and WF_{sc}(u) ⊂ Φ_{+}({q}).

Proof. Suppose first that q ∈ Min_{+}(λ) then any microlocal eigenfunction, u ∈ Ẽ_{mic,+}(O, λ), has q as (at most) the only point in its wavefront set within O. Using a microlocal cutoff around q with support in O gives a map which realizes the isomorphism in (5.4).

If q ∈ Max_{+}(λ) then ũ constructed in the proof of Lemma 5.6 provides a representative such that (P − λ)u ∈ Ċ^{∞}(X), with WF_{sc}(ũ) contained in the union of Φ_{+}({q}) and Φ_{+}({q′}), for q′ ∈ RP_{+}(λ) with ν(q′) > ν(q). If we choose q′ from the set

(5.5)  {q′ ∈ RP_{+}(λ) ∩ WF_{sc}(ũ) | ν(q′) > ν(q), q′ ∉ Φ_{+}({q})},

with ν(q′) minimal, then by localizing ũ near q′ we have an element v of E_{mic,+}(q′). By subtracting from ũ a representative for v given by Lemma 5.6, we remove the wavefront set near q′. Inductively choosing radial points from (5.5) and performing this procedure, we remove all wavefront set from ũ except that contained in Φ_{+}({q}). □

At radial points corresponding to maxima these results can be strengthened by using a positive commutator estimate which is dual, in some sense, to Proposition 4.13.

Proposition 5.8. Assume that q ∈ RP_{+}(λ), and let O be a W -balanced neighbourhood of q. If, for some r < −1/2, O ∩ WF_{sc}^{∗,r+1}((P − λ)u) = ∅ and O ∩ WF_{sc}^{∗,r}(u) ⊂ Φ_{+}({q}), then O ∩ WF_{sc}^{∗,r}(u) = ∅. The same holds true if q ∈ RP_{−}(λ), u ∈ E_{mic,−}(q, λ), and O ∩ WF_{sc}^{∗,r}(u) ⊂ Φ_{−}({q}).

Proof. Assume that q ∈ Max_{+}(λ) for the sake of definiteness, and let L_{+} be that one of the Legendrians L_{1}, L_{2} from Proposition 1.5 with L_{+} ⊂ {ν ≥ ν(q)}. The argument for q ∈ Min_{+}(λ) is similar, and slightly easier since in that case Φ_{+}({q}) consists of the single point q. The case q ∈ RP_{−}(λ) follows by taking the complex conjugate.

Let l < −1, l < r, be such that u ∈ H_{sc}^{∗,l}(X), let ν_{0} = ν(q) > 0 and let y_{0} = y(q) in some local coordinates. Let a, χ, ϕ, ψ̃ and ψ_{0} be as in Proposition 4.13 (except that here ν_{0} > 0). Then we may write

(5.6)  ^{sc}H_{p}a = −b^{2}x^{−l−1} + ex^{−l−1}

as in Proposition 4.13, since now l+1 has changed signs, but so has ν (from negative to positive) on the support of χ. If we use Lemma 4.3 instead of Lemma 4.1 (since we are dealing here with a single function rather than a family), the argument from Proposition 4.13 applies, except we can only reach values of r less than −1/2 since we must have l < −1. □

Corollary 5.9. If q ∈ RP_{+}(λ), O is a W -balanced neighbourhood of q and u ∈ Ẽ_{mic,+}(O, λ), then WF_{sc}^{∗,r}(u) ∩ O = ∅ for all r < −1/2.

6. Test modules and iterative regularity

To investigate the regularity of microlocal eigenfunctions we use ‘test modules’ of scattering pseudodifferential operators. Since we work microlocally we consider scattering pseudodifferential operators microlocally supported in an open set O ⊂ C = ∂(^{sc}\overline{T}^{∗}X). In particular we shall use the notation

(6.1)  Ψ_{sc}^{m,l}(O) = {A ∈ Ψ_{sc}^{m,l}(X); WF′_{sc}(A) ⊂ O} .

Since we are only interested in boundary regularity here we shall suppose that O ⊂ C_{∂} = ^{sc}T_{∂X}^{∗}X (see (1.5)) and so we may take m = −∞.

Definition 6.1. A test module in an open set O ⊂ C_{∂} is a linear subspace M ⊂ Ψ_{sc}^{−∞,−1}(O) which contains and is a module over Ψ_{sc}^{−∞,0}(O), which is closed under commutators and which is finitely generated in the sense that there exist finitely many A_{i} ∈ Ψ_{sc}^{∗,−1}(X), i = 0, 1, …, N, A_{0} = Id, such that each A ∈ M can be written

(6.2)  A = ∑_{j=0}^{N} Q_{i}A_{i}, Q_{i} ∈ Ψ_{sc}^{−∞,0}(O).

Since we have assumed that M ⊂ Ψ_{sc}^{−∞,−1}(X), A ∈ M, C ∈ Ψ_{sc}^{−∞,0}(X) implies that [A, C] ∈ Ψ^{−∞,0}(O) ⊂ M. The structure of the module is thus determined by the commutators of the generators. It certainly suffices to have

(6.3) [A_{i}, A_{j}] = ∑_{k=0}^{N} C_{ijk}A_{k} + E′_{jk}, C_{ijk} ∈ Ψ_{sc}^{0,0}(X),

E′_{jk} ∈ Ψ_{sc}^{0,−1}(X), WF′_{sc}(E′_{jk}) ∩ O = ∅.

This in turn follows from the purely symbolic condition on principal symbols:

(6.4)  \{a_{i}, a_{j}\} = ∑_{k=0}^{N} c_{ijk}a_{k} + e_{ij} in a neighbourhood of \overline{O}

where the e_{ij} are symbols of order 0.

In our arguments below, we frequently shrink O, so the condition (6.3) is in essence necessary. When O′ ⊂ O is another open set we consider the restricted module over O′

(6.5)  M(O′) = ∑_{i=0}^{N} Ψ^{−∞,0}(O′) · A_{i}.

which is easily seen to be independent of the choice of generators.

Definition 6.2. Let M be a test module. For u ∈ C^{−∞}(X) we say u ∈ I_{sc}^{(s)}(O, M) if M^{m}u ⊂ H_{sc}^{∞,s}(X) for all m. That is, for all m and for all B_{i} ∈ M, i = 1, …, m, we have ∏_{i=1}^{m} B_{i}u ∈ H_{sc}^{∞,s}(X). We may define the finite regularity spaces by saying u ∈ I_{sc}^{(s),M}(O, M) if M^{m}u ⊂ H_{sc}^{∞,s}(X) for all m ≤ M.

Note that I_{sc}^{(s)}(O, M) is a well-defined space of microfunctions over O since if u ∈ C^{−∞}(X) and WF_{sc}(u) ∩ O = ∅ then Au ∈ Ċ^{∞}(X) for all A ∈ Ψ_{sc}^{∗,∗}(X) such that WF′_{sc}(A) ⊂ O; in particular this is the case for all A ∈ M. Clearly

(6.6)  I_{sc}^{(s)}(O, M) = ⋂_{M=0}^{∞} I_{sc}^{(s),M}(O, M).

From the definition of the restriction in (6.5) it also follows that if O′ ⊂ O then

(6.7)  I_{sc}^{(s),k}(O, M) ⊂ I_{sc}^{(s),k}(O′, M).

Since Id ∈ M,

Ψ_{sc}^{−∞,0}(O) = M^{0} ⊂ M ⊂ M^{2} ⊂ ….

Definition 6.2 above then reduces to

(6.8)  u ∈ I_{sc}^{(s),M}(O, M) ⟺ M^{M}(O)u ⊂ H_{sc}^{∞,s}(X).

If two test modules M_{1} and M_{2} define equivalent filtrations of the same set of operators they define the same space of iteratively-regular distributions. That is,

(6.9) M_{1}(O) ⊂ (M_{2}(O))^{k_{1}} and M_{2}(O) ⊂ (M_{1}(O))^{k_{2}} ⟹

I_{sc}^{(s)}(O, M_{1}) = I_{sc}^{(s)}(O, M_{2}), ∀ s ∈ ℝ.

Lemma 6.3. If A_{i}, 0 ≤ i ≤ N, are generators for M in the sense of Definition 6.2 with A_{0} = Id, then

(6.10)  M^{k} = { ∑_{|α|≤k} Q_{α} ∏_{i=1}^{N} A_{i}^{α_{i}}, Q_{α} ∈ Ψ^{−∞,0}(O)}

where α runs over multiindices α : {1, …, N} ⟶ ℕ_{0} and |α| = α_{1} + ⋯ + α_{N}.

Proof. This is just a consequence of the fact that M is a Lie algebra and module. Thus, we proceed by induction over k since (6.10) certainly holds for k = 1, and by definition for k = 0. It therefore suffice to prove the equality (6.10) modulo M^{k−1}. Then (6.10) just corresponds to [M^{r}, M^{s}] ⊂ M^{r+s−1}, applied with r + s ≤ k, which allows the factors to be freely rearranged. □

In the modules we consider, one of the generating elements is A_{N} = x^{−1}(P − λ) and we deal with approximate eigenfunctions, so that typically (P −λ)u ∈ Ċ^{∞}(X). We may reorder the basis so that this element comes last. In this case it is enough to consider the action on u of the remaining generators in an inductive proof of regularity since if an operator as on the right hand side of (6.10) with α_{N} ≠ 0 is applied to u then we obtain an element of Ċ^{∞}(X). We therefore use the ‘reduced’ multiindex notation, with an additional index to denote the weight, and set

(6.11)  A_{α,s} = x^{−s} ∏_{i=1}^{N−1} A_{i}^{α_{i}}, α_{i} ∈ ℕ_{0}, 1 ≤ i ≤ N − 1.

Corollary 6.4. Suppose M is a test module, A_{N} = x^{−1}(P − λ) is a generator, u ∈ C^{−∞} satisfies (P − λ)u ∈ Ċ^{∞}(X) and u ∈ I_{sc}^{(s),m−1}(O, M). Then for O′ ⊂ O, u ∈ I_{sc}^{(s),m}(O′, M) if for each multiindex α, with |α| = m, α_{N} = 0, there exists Q_{α} ∈ Ψ^{−∞,0}(X), elliptic on O′ such that Q_{α}A_{α,s}u ∈ L_{sc}^{2}(X).

In the case of interest here, O will be a W -balanced neighbourhood of q ∈ RP(λ) and the generators A_{i} will be certain operators, related to the geometry of the classical dynamics near q, with principal symbols vanishing at q. The last operator, as mentioned above, will be A_{N} = x^{−1}(P −λ). Our main tool in the demonstration of regularity with respect to such a test module is the use of positive commutator techniques. These depend on finer structure of the module. In particular the basic source of positivity is the commutation relation

(6.12) i[x^{−s− \frac{1}{2}} , P − λ] = x^{−s+ \frac{1}{2}} C_{0},

where C_{0} ∈ Ψ_{sc}^{1,0}(X) has principal symbol − (2s + 1)ν,

so has a definite sign in a neighbourhood of a radial point q ∈ RP_{±}(λ) if s ≠ −1/2 and λ ∉ Cv(V). This strict sign of σ_{0}(C_{0}) allows us to deal with error terms. To fix signs we shall assume that q ∈ RP_{+}(λ) and λ ∉ Cv(V) in the remainder of this section.

So suppose that f(ν, y, μ) vanishes at q ∈ RP_{+}(λ), and df is an eigenvector for the linearization of W at q with eigenvalue −2νr, where ν = ν(q) > 0. Then

(6.13)  ^{sc}H_{p}(x^{m}f) = x^{m}[−2ν(m + r)f + f_{2} + O(x)],

where f_{2} is smooth and vanishes quadratically at the critical point q. Thus,

scH_{p}(x^{2m}f^{2}) = 2x^{2m}[−2ν(m + r)f^{2} + f_{2}f + O(x)].

If m + r ≠ 0 and f_{2} is of the form ff̃_{2}, then this has a fixed sign near q. In general, f_{2} does not factor this way, so we need to consider a sum of such commutators corresponding to functions f, f′ such that df and df′ span T_{q}^{∗}Σ(λ); then x^{−2m} ^{sc}H_{p}(x^{2m}f^{2}) + x^{−2m} ^{sc}H_{p}(x^{2m}(f′)^{2}) has a fixed sign near q on Σ(λ). We use this as a basis for the construction of positive commutators, though it is more convenient to describe our conditions on the generators A_{i}, and then spell out in detail in the following sections why these conditions hold.

The basic condition we require is that the remaining generators of the test module, A_{i}, i = 1, …, l = N − 1, satisfy

(6.14)

x^{−1}i[A_{i}, P − λ] = ∑_{j=0}^{N} C_{ij}A_{j}, C_{ij} ∈ Ψ_{sc}^{∗,0}(X),

where σ_{∂}(C_{ij})(q) = 0, j ≠ i,

and Re σ_{∂}(C_{jj})(q) ≥ 0, 0 < j ≤ l, q ∈ RP_{+}(λ), s < −\frac{1}{2}.

However, we will occasionally need slightly weaker conditions than (6.14). These weaker conditions allow σ_{∂}(C_{ij})(q) ≠ 0 even for i ≠ j, but they require that the matrix σ_{∂}(C_{ij})(q) is lower triangular. Hence, in this case, the ordering of the A_{i} (by their index) is not arbitrary. Our weaker condition is then

(6.15)

x^{−1}i[A_{i}, P − λ] = ∑_{j=0}^{N} C_{ij}A_{j}, C_{ij} ∈ Ψ_{sc}^{∗,0}(X),

where σ_{∂}(C_{ij})(q) = 0, j > i,

and Re σ_{∂}(C_{jj})(q) ≥ 0, 0 < j ≤ l, q ∈ RP_{+}(λ), s < −\frac{1}{2}..

We need the following notation. Suppose that α, β are multiindices. We say that α ∼ β if there exist i ≠ j such that for k ∉ {i, j}, α_{k} = β_{k}, while α_{i} = β_{i} + 1, α_{j} = β_{j} − 1. This means that A_{α} has an additional factor of A_{i}, and one fewer factor of A_{j} than A_{β}. If α ∼ β, we write α < β provided that i < j.

The following is a technical Lemma we need for Proposition 6.8 below, which gives a method for showing membership of u ∈ I_{sc}^{(s),m}(O, M).

Lemma 6.5. Let q ∈ RP_{+}(λ) and s < −\frac{1}{2}, where λ ∉ Cv(V). Suppose Q ∈ Ψ_{sc}^{−∞,0}(O), let K_{α} > 0 be arbitrary positive constants, and let C_{0} and C_{ij} be given by (6.12), resp. (6.14). Then, assuming (6.14), and using the notation of (6.11), we have

(6.16) ∑_{|α|=m} iK_{α}[A_{α,s+1/2}^{∗}Q^{∗}QA_{α,s+1/2}, P − λ]

= ∑_{|α|,|β|=m} A_{α,s}^{∗}Q^{∗}C′_{αβ}QA_{β,s}

+ ∑_{|α|=m} (A_{α,s}^{∗}Q^{∗}E_{α,s} + E_{α,s}^{∗}QA_{α,s}) + ∑_{|α|=m} A_{α,s}^{∗}i[Q^{∗}Q, P − λ]A_{α,s},

where

(6.17) E_{α,s} = x^{−s}E_{α}, E_{α} ∈ M^{m−1} + M^{m−1}A_{N}, WF′_{sc}(E_{α}) ⊂ WF′_{sc}(Q),

C′_{αβ} ∈ Ψ_{sc}^{1,0}(X), σ_{∂}(C′_{αα})(q) = 2K_{α} (−(2s + 1)ν + ∑_{j} α_{j} Re σ_{∂}(C_{jj})(q)) > 0

and σ_{∂}(C′_{αβ})(q) = 0 for α ≠ β.

(6.18)

In addition, C′_{αβ} = 0 unless either α = β or α ∼ β.

Assuming instead (6.15) instead of (6.14), the conclusions hold with the last equation of (6.18) replaced by

\begin{matrix}σ_{∂}(C′_{αβ})(q) = K_{β}β_{j}\overline{σ_{∂}(C_{ji})(q)} for α ∼ β, α < β, \\ σ_{∂}(C′_{αβ})(q) = K_{α}α_{i}σ_{∂}(C_{ij})(q) for α ∼ β, α > β. \end{matrix} ; (6.19)

Remark 6.6. The first term on the right hand side of (6.16) is the principal term, in terms of M-order, since both the A_{α,s}^{∗} and A_{β,s} terms have M-order equal to m. Note that this term has nonnegative principal symbol. In the second term, the E terms have M-order m − 1 which allows these to be treated as error terms. To deal with the third term, involving the commutator [Q^{∗}Q, P − λ], we need additional information about Q, discussed below.

Proof. We obtain the identity (6.16) by computing

(6.20) [QA_{α,s+1/2}, P − λ] = [Qx^{−s−1/2}A_{1}^{α_{1}} … A_{N−1}^{α_{N−1}}, P − λ],

Q ∈ Ψ_{sc}^{−∞,0}(O), |α| = m.

The commutator with P − λ distributes over the product, in each term replacing a single factor A_{i} by ∑ xC_{ij}A_{j}. After rearrangement of the order of the factors, giving an error included in E_{α} below, it becomes

(6.21) (C_{0} + ∑_{j} α_{j}C_{jj})Qx^{−s+1/2}A_{α} + ∑_{|β|=m, β≠α} C_{αβ}Qx^{−s+1/2}A_{β}

+ x^{−s+1/2}E_{α} + i[Q, P − λ]A_{α,s+1/2},

where σ_{∂}(C_{αβ})(q) = 0, C_{αβ} = 0 unless α = β or α ∼ β, and the E_{α} are as in (6.17)

– using in particular the inclusion xM^{m} ⊂ M^{m−1}. In addition, for α ∼ β,

σ_{∂}(C_{αβ})(q) = α_{i}σ_{∂}(C_{ij})(q).

This implies that

(6.22) i[A_{α,s+1/2}^{∗}Q^{∗}QA_{α,s+1/2}, P − λ]

= A_{α,s}^{∗}Q^{∗}(C_{0} + C_{0}^{∗} + ∑_{j} α_{j}(C_{jj} + C_{jj}^{∗}))QA_{α,s}

+ ∑_{β} A_{α,s}^{∗}Q^{∗}C_{αβ}QA_{β,s} + ∑_{β} A_{β,s}^{∗}Q^{∗}C_{αβ}^{∗}QA_{α,s} + A_{α,s}^{∗}Q^{∗}E_{α,s}

+ E_{α,s}^{∗}QA_{α,s} + A_{α,s}^{∗}i[Q^{∗}Q, P − λ]A_{α,s};

here E_{α,s} = x^{−s}E_{α} and all sums over β are understood as |β| = m, β ≠ α. Let

\begin{matrix}C′_{αα} = K_{α} (C_{0} + C_{0}^{∗} + ∑_{j} α_{j}(C_{jj} + C_{jj}^{∗})) \\ C′_{αβ} = K_{α}α_{i}C_{ij} + K_{β}β_{j}C_{ji}^{∗} if α ∼ β. \end{matrix} ; (6.23)

If we assume (6.14), we deduce that (6.18) holds, while if we assume (6.15), (6.19) follows. This finishes the proof. □

We now consider C′ = (C′_{αβ}), |α| = m = |β| as a matrix of operators, or rather as an operator on a trivial vector bundle with fiber ℝ^{|M_{m}|} over a neighbourhood of z = π(q) in X, where |M_{m}| denotes the number of elements of the set M_{m} of multiindices α with |α| = m. Let c′ = σ_{∂}(C′)(q), c′_{αβ} = σ_{∂}(C′_{αβ})(q).

Suppose first that (6.14) holds. Then for any choice of K_{α} > 0, e.g. K_{α} = 1 for all α, c′ = σ_{∂}(C′)(q) is positive or negative definite with the sign of σ_{∂}(C_{0})(q) (whose sign for q ∈ RP_{+}(λ) is that of −s − 1/2). The same is therefore true microlocally near q. For the sake of definiteness, suppose that σ_{∂}(C_{0})(q) > 0. Then there exist a neighbourhood O_{m} of q, depending on |α| = m, and B ∈ Ψ_{sc}^{−∞,0}(X), G ∈ Ψ_{sc}^{−∞,1}(X), with σ_{∂}(B) > 0 on O_{m} such that

(6.24)  Q ∈ Ψ_{sc}^{−∞,0}(X), WF′_{sc}(Q) ⊂ O_{m} ⇒ Q^{∗}C′Q = Q^{∗}(B^{∗}B + G)Q.

Assume now that (6.15) holds. There is a natural partial order on M_{m} given as follows: α < β if there exist α^{(j)}, j = 0, 1, …, n, n ≥ 1, α^{(0)} = α, α^{(n)} = β, α^{(j)} ∼ α^{(j+1)} and α^{(j)} < α^{(j+1)} (which has already been defined if α^{(j)} ∼ α^{(j+1)}) for j = 0, 1, …, n − 1. This is clearly transitive, and if α < β then ∑_{j} jα_{j} < ∑_{j} jβ_{j}, so α < β and β < α cannot hold at the same time. We place a total order <′ on M_{m} which is compatible with <, so α < β implies α <′ β. For the sake of definiteness, suppose that σ_{∂}(C_{0})(q) > 0. We then choose K_{α} inductively starting at the maximal element of <′, making σ_{∂}(C′)(q) positive definite on ⊕_{α≤′β}ℝ_{β}. We can for example take K_{α} = 1 for the maximal element α. Once K_{β} has been chosen for all β with α <′ β, so that c′ is positive definite on ⊕_{α<′β}ℝ_{β}, we remark that of all entries c′_{βγ} with α ≤ β, γ, only c′_{αα} depends on K_{α}. Expanding (v, c′v) for v = (v′, v′′) ∈ ℝ_{α} ⊕ (⊕_{α<′β}ℝ_{β}) , and using Cauchy-Schwarz for the cross terms involving both v′ and v′′, we deduce that for any ϵ > 0,

(v, c′v) ≥ c′_{αα}∥v′∥^{2} + (v′′, c′v′′) − ϵ∥c′v′′∥^{2} − ϵ^{−1}∥v′∥^{2}.

Since c′ is positive definite on ⊕_{α<′β}ℝ_{β}, we can choose first ϵ > 0 sufficiently small, then K_{α} > 0 sufficiently large (depending on ϵ) such that c′ is positive definite on ⊕_{α≤′β}ℝ_{β}. Thus, for suitable constants K_{α}, c′ = σ_{∂}(C′)(q) is positive definite, and we again conclude that there exist a neighbourhood O_{m} of q, depending on |α| = m, and B ∈ Ψ_{sc}^{−∞,0}(X), G ∈ Ψ_{sc}^{−∞,1}(X), with σ_{∂}(B) > 0 on O_{m} such that

(6.25)  Q ∈ Ψ_{sc}^{−∞,0}(X), WF′_{sc}(Q) ⊂ O_{m} ⇒ Q^{∗}C′Q = Q^{∗}(B^{∗}B + G)Q.

Equations (6.12) and (6.14) (or (6.12) and (6.15)) are all that is needed in the two cases of centers and sinks, where we know a priori that WF_{sc}(u) ⊂ {q}. In these cases we take Q such that q ∉ WF′_{sc}(Id −Q), which implies that

(6.26)  i[Q^{∗}Q, P − λ] = x^{1/2}F̃x^{1/2}, where F̃ ∈ Ψ_{sc}^{0,0}(O), q ∉ WF′_{sc}(F̃).

In the case of a saddle point, i.e. q ∈ Max_{+}(λ), we need a more general setup and we shall arrange that q ∉ WF′_{sc}(Id −Q),

i[Q^{∗}Q, P − λ] = x^{1/2}(B̃^{∗}B̃ + G̃)x^{1/2} + x^{1/2}F̃x^{1/2}, where
B̃, F̃ ∈ Ψ_{sc}^{0,0}(O), G̃ ∈ Ψ_{sc}^{0,1}(X), q ∉ WF′_{sc}(F̃),

(6.27)

and in addition, F̃ satisfies WF′_{sc}(F̃) ⊂ {ν < ν(q)}. This condition on F̃ will ensure that WF′_{sc}(F̃) ∩ WF_{sc}(u) = ∅ for the application in section 9.

Proposition 6.7. Suppose that m > 0, s < −1/2, q ∈ RP_{+}(λ), λ ∉ Cv(V), either (6.14) or (6.15) hold, and let O_{m} be as in (6.24) (or (6.25)). Suppose that u ∈ I_{sc}^{(s),m−1}(O_{m}, M), WF_{sc}((P − λ)u) ∩ O_{m} = ∅ and that there exists Q ∈ Ψ_{sc}^{−∞,0}(O_{m}) elliptic at q that satisfies (6.27) with WF′_{sc}(F̃) ∩ WF_{sc}(u) = ∅. Then u ∈ I_{sc}^{(s),m}(O′, M) where O′ is the elliptic set of Q.

Proof. First consider u′ ∈ I_{sc}^{(s),m}(O_{m}, M). Let Au′ = (QA_{α,s}u′)_{|α|=m}, regarded as a column vector of length |M_{m}|. Now consider

\begin{matrix}∑_{|α|=m} K_{α}⟨u′, i[A_{α,s+1/2}^{∗}Q^{∗}QA_{α,s+1/2}, P − λ]u′⟩ \\ = ∥BAu′∥^{2} + ⟨Au′, GAu′⟩ + ∑_{|α|=m} (⟨QA_{α,s}u′, E_{α,s}u′⟩ + ⟨E_{α,s}u′, QA_{α,s}u′⟩) \\ + ∑_{|α|=m} (∥B̃A_{α,s}u′∥^{2} + ⟨A_{α,s}u′, F̃A_{α,s}u′⟩ + ⟨A_{α,s}u′, G̃A_{α,s}u′⟩). \end{matrix} (6.28)

Hence, dropping the term involving B̃ and applying the Cauchy-Schwarz inequality to the terms with E_{α,s}, G and G̃, we have for any ϵ > 0,

(6.29) ∥BAu′∥^{2} ≤ ∑_{α} |⟨u′, i[A_{α,s+1/2}^{∗}Q^{∗}QA_{α,s+1/2}, P − λ]u′⟩|

+ ϵ(∥Au′∥^{2} + ∑_{α}(∥QA_{α,s}u′∥^{2} + ∥A_{α,s}u′∥^{2}))

+ ϵ^{−1}(∥GAu′∥^{2} + ∑_{α}(∥E_{α,s}u′∥^{2} + ∥K_{α}G̃A_{α,s}u′∥^{2})) + |⟨A_{α,s}u′, F̃A_{α,s}u′⟩|.

Choosing ϵ > 0 small enough, the second term on the right can be absorbed in the left hand side (since B is strictly positive), and we get

\begin{aligned}\frac{1}{2}∥BAu′∥^{2} ≤ & ∑_{α} |⟨u′, i[A_{α,s+1/2}^{∗}Q^{∗}QA_{α,s+1/2}, P − λ]u′⟩| \\ & + ϵ^{−1}(∥GAu′∥^{2} + ∑_{α}(∥E_{α,s}u′∥^{2} + ∥K_{α}G̃A_{α,s}u′∥^{2})) \\ & + |⟨A_{α,s}u′, F̃A_{α,s}u′⟩|. \end{aligned} ; (6.30)

We apply this with u′ replaced by u_{r} = (1 + r/x)^{−1}u = \frac{x}{x+r} u r > 0, where

now u ∈ I_{sc}^{(s),m−1}(O_{m}, M). Then letting r → 0, using the strong convergence of (1 + r/x)^{−1} to the identity as in the argument of Lemma 4.3, and the assumption that WF′_{sc}(F̃) ∩ WF_{sc}(u) = ∅, shows that BAu ∈ L_{sc}^{2}(X), finishing the proof. □

We may arrange that for all m,

(6.31)  \overline{O_{m+1}} ⊂ O_{m} and Q = Q_{m} is elliptic on O_{m+1}.

By induction, the proposition implies that if WF_{sc}^{∗,s}(u)∩O_{1} = ∅ for some s < −1/2, and WF_{sc}((P − λ)u) ∩ O_{1} = ∅ then for all m, u ∈ I_{sc}^{(s),m}(O_{m+1}). Since we may have ∩_{m=1}^{∞}O_{m} = {q}, we need an additional assumption, amounting to ‘propagation of regularity’, to extend the result to I_{sc}^{(s)}(O, M) for some neighbourhood O of q. Of course, there is no need for such an assumption if WF_{sc}(u) ∩ O ⊂ {q} for then u ∈ I_{sc}^{(s),m}(O_{m+1}, M) implies that u ∈ I_{sc}^{(s),m}(O, M). We state these two cases separately, but prove them together.

Proposition 6.8. Suppose that s < −1/2, q ∈ RP_{+}(λ), λ ∉ Cv(V), either (6.14) or (6.15) hold, and let O be a neighbourhood of q. Then for any u ∈ C^{−∞}(X),

WF_{sc}^{∗,s}(u) ∩ O = ∅, WF_{sc}(u) ∩ O ⊂ {q} and WF_{sc}((P − λ)u) ∩ O = ∅
⇒ u ∈ I_{sc}^{(s)}(O, M).

In the next proposition we assume ‘propagation of regularity’ to show that u ∈ I_{sc}^{(s),m}(O, M).

Proposition 6.9. Suppose that s < −1/2, q ∈ RP_{+}(λ), λ ∉ Cv(V), either (6.14) or (6.15) hold, and let Q and O_{m} be as in (6.24) (or (6.25)), (6.27) and (6.31). Suppose also that there exists a neighbourhood O of q such that for all m,

(6.32) u ∈ I_{sc}^{(s),m}(O_{m+1}, M) and WF_{sc}((P − λ)u) ∩ O = ∅ ⇒ u ∈ I_{sc}^{(s),m}(O, M).

Then for any u ∈ C^{−∞}(X), satisfying

WF_{sc}^{∗,s}(u) ∩ O = ∅ and WF_{sc}((P − λ)u) ∩ O = ∅ ⇒ u ∈ I_{sc}^{(s)}(O, M).

Proof. By hypothesis, u ∈ I_{sc}^{(s),0}(O, M). We now show that for m ≥ 1, u ∈ I_{sc}^{(s),m−1}(O, M) implies that u ∈ I_{sc}^{(s),m}(O, M). But indeed, u ∈ I_{sc}^{(s),m−1}(O, M) implies that u ∈ I_{sc}^{(s),m−1}(O_{m}, M), hence u ∈ I_{sc}^{(s),m}(O_{m+1}, M) by Proposition 6.7 and (6.31). In Proposition 6.8, WF_{sc}(u) ∩ O ⊂ {q}, so u ∈ I_{sc}^{(s),m}(O_{m+1}, M) implies that u ∈ I_{sc}^{(s),m}(O, M). In Proposition 6.9, by (6.32), u ∈ I_{sc}^{(s),m}(O, M). This completes the inductive step. □

Remark 6.10. If q is a sink or center, we use Proposition 6.8. If q is a saddle, we use Proposition 6.9. In that case, we take O to be a W -balanced neighbourhood of q. Then u ∈ I_{sc}^{(s),m}(O, M) is a finite Legendre regularity condition, and (6.32) holds by propagation of Legendre regularity outside critical points of the bicharacteristic flow, which is an immediate consequence of the parametrix construction of Duistermaat and Hörmander. The propagation of Legendre regularity away from critical points can also be proved by the use of test modules as in this section. In fact, the use of (6.32) can be avoided altogether if we construct Q more carefully and use the term with B̃ as well in (6.28). Namely, fix a small neighbourhood O′ of q. Next, given m, we first choose K_{α} > 0 so that C′ is positive definite at q. Then we construct Q such that q ∉ WF_{sc}(Id −Q), Q elliptic on O′ and (6.27) holds, and in addition (B̃^{∗}B̃)_{|α|=m} + Q^{∗}C′Q (the first term denoting a diagonal matrix) can be written as B^{∗}B + F + G with WF′_{sc}(F) ∩ WF_{sc}(u) = ∅, G ∈ Ψ_{sc}^{−∞,1}(X). We can do this since for any constant N = N_{m}, we can arrange that Nq_{0} ≤ − ^{sc}H_{p}q_{0} on a neighborhood of WF_{sc}(u) outside a small m-dependent neighborhood of q where q_{0} = σ_{∂}(Q). Then the proof of Proposition 6.7 applies directly and shows that u ∈ I_{sc}^{(s),m}(O′, M) for all m.

7. Outgoing eigenfunctions at a center

We first analyze the structure of elements of E_{mic,+}(q, λ) when q ∈ Min_{+}(λ) is a center for the vector field W, i.e. when

(7.1)  2V′′(z) > λ − V (z), z = π(q).

As shown in Corollary 5.7, by microlocalizing the solution near q we may simply assume that

(7.2)  (P − λ)u = f ∈ Ċ^{∞}(X), u ∈ C^{−∞}(X), WF_{sc}(u) ⊂ {q}.

We proceed in three steps, first using the commutator methods outlined above to obtain iterative regularity, with respect to the Ψ_{sc}^{−∞,0}(O)-module (7.3)

M generated by Id, \{x^{− \frac{1}{2}} A, A ∈ Ψ_{sc}^{−∞,0}(O), σ_{∂}(A)(q) = 0\} and x^{−1}(P − λ).

Then we pass to a blown up space where this regularity becomes conormal regularity, once a phase is factored out. Finally, using the equation again, we find the expansion (I.20).

Proposition 7.1. The module M in (7.3) is a test module in the sense of Definition 6.2, which satisfies (6.14). If u satisfies (7.2) at q, then u ∈ I_{sc}^{(−1/2−ϵ)}(O, M) for all ϵ > 0.

Proof. Choose a complex symbol a ∈ C_{c}^{∞}(^{sc}T^{∗}X) such that da_{j}(q), restricted to Σ(λ) at q, is a non-trivial eigenvector for the linearization of W with eigenvalue −2ν̃r_{1} ∈ ℂ; here ν̃ = ν(q). Then ā has the same property with eigenvalue −2ν̃r_{2} = −2ν̃\overline{r_{1}} and together they generate, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions vanishing at q. Recall from (1.25) that Re r_{1} = Re r_{2} = 1/2. Certainly A ∈ x^{− \frac{1}{2}} Ψ_{sc}^{−∞,0}(X) with symbol x^{− \frac{1}{2}} a is in the set in (7.3) and indeed M is generated by A_{0} = Id, A_{1} = A, A_{2} = A^{∗} and A_{3} = x^{−1}(P − λ).

By (6.13), the commutator i[A_{j}, P − λ] ∈ Ψ_{sc}^{−∞,−1/2}(X) has principal symbol

(7.4)  − ^{sc}H_{p}(x^{−1/2}a_{j}) = 2ν̃x^{−1/2}(i Im(r_{j})a_{j} + e_{j}), j = 1, 2,

where the e_{j} vanish to second order at q. Thus

(7.5)  ix^{−1}[A_{j}, P − λ] = C_{j0}x^{1/2} + C_{j1}A_{1} + C_{j2}A_{2} + x^{1/2}C_{j3}A_{3}, j = 1, 2,

with C_{jk} ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(C_{jk})(q) = 0 if j ≠ k, and where σ_{∂}(C_{jj})(q) = i2ν̃ Im(r_{j}) is pure imaginary. In particular M is closed under commutators and hence is a test module satisfying (6.14).

To prove the statement about u, note that WF_{sc}^{∗,−1/2−ϵ}(u) ∩ O = ∅ for any W - balanced neighbourhood O of q, by Corollary 5.9. If we take any Q ∈ Ψ_{sc}^{−∞,0}(O) with q ∉ WF′_{sc}(Id −Q), then (6.26) is satisfied, so Proposition 6.8 yields the result.

□

Remark 7.2. The factor x^{−1/2} in (7.3) is chosen precisely that the real part of r_{j} in (7.4) is cancelled, allowing the application of Proposition 6.8. A similar cancellation is crucial in (8.2) in the next section.

Next we reinterpret this iterative regularity more geometrically by introducing the parabolically blown up space

(7.6)  X_{z} = [X; {z}]_{1/2}, β : X_{z} ⟶ X.

Such parabolic blow-ups are defined in considerable generality, including this case, in [2]. We recall a form of the construction, sufficient for our purposes here, in Section 2. Recall that X_{z} is a compact manifold with corners, of dimension two. The two boundary curves are the front face ff = β^{−1}(z), created by the blowup, and the closure of the pullback of ∂X \ {z} which we shall refer to as the old boundary.

We shall denote boundary defining functions for these boundary curves by ρ_{ff} and ρ_{o}, respectively. We also have boundary defining functions for the other boundary curves of X, which lift unchanged to boundary curves of X_{z}; we shall denote the product of these as ρ′ since they play no significant rôle here.

One class of natural Sobolev spaces on such a manifold with corners is the class of weighted b-Sobolev spaces, which we proceed to define. Let V_{b}(X_{z}) be the space of smooth vector fields on X_{z} which are tangent to all boundary faces and let μ_{b} be a b-density on X_{z}, i.e. , a density of the form (ρ′ρ_{o}ρ_{ff})^{−1}μ, where μ is a smooth, nonvanishing density. Then we set

(ρ′)^{a}ρ_{o}^{b}ρ_{ff}^{c}H_{b}^{M}(X_{z}) = {u; u = (ρ′)^{a}ρ_{o}^{b}ρ_{ff}^{c}v, v ∈ H_{b}^{M}(X_{z})} ,

H_{b}^{M}(X_{z}) = {u ∈ L^{2}(X_{z}; μ_{b}); V_{b}^{k}u ⊂ L^{2}(X_{z}; μ_{b}) ∀ k ≤ M} . (7.7)

These are the conormal functions (conormal with respect to the boundary) and we also use the short-hand notation

(7.8)  (ρ′)^{∞}ρ_{o}^{∞}ρ_{ff}^{c}H_{b}^{∞}(X_{z}) = ⋂_{a,b}(ρ′)^{a}ρ_{o}^{b}ρ_{ff}^{c}H_{b}^{∞}(X_{z})

for the subspaces which are rapidly decreasing up to the boundaries other than ff .

Proposition 7.3. If q is a center for W in Σ(λ), with M the test module given by (7.3) in a W -balanced neighbourhood of q then, for any s, multiplication gives an isomorphism

(7.9) {u ∈ C^{−∞}(X); WF_{sc}(u) ⊂ {q}, u ∈ I^{(s)}(O, M)} ∋ u ⟼

e^{−iν(q)/x}u ∈ (ρ′)^{∞}ρ_{o}^{∞}ρ_{ff}^{2s+3/2}H_{b}^{∞}(X_{z}).

Proof. We shall work in Riemannian normal coordinates in the boundary based at the minimum under discussion. In terms of these coordinates the boundary symbol of P − λ is

(7.10) p − λ = ν^{2} + μ^{2} + V_{0}(0) + y^{2}a(y) − λ + O(x), y^{2}a(y) = V_{0}(y) − V_{0}(0).

Here μ = 0, y = 0 and ν = ν̃ = \sqrt{λ − V_{0}(0)} > 0 and a(0) = V′′_{0}(0)/2 > 0 at the radial point. The module M is therefore generated by Id, x^{\frac{1}{2}} D_{y}, x^{− \frac{1}{2}} y and

(7.11)  x^{−1} ((x^{2}D_{x})^{2} + x^{2}D_{y}^{2} − ν̃^{2} + y^{2}a(y)) .

Consider the effect of conjugation by e^{iν̃/x}. This maps each Ψ_{sc}^{m,l}(X) isomorphically to itself and hence

(7.12)  M̃ = e^{−iν̃/x}Me^{iν̃/x}

is another test module, but in the open set

(7.13)  Õ = {(ν − ν̃, y, μ); (ν, y, μ) ∈ O} ∋ q̃ = (0, 0, 0),

q̃ being the image of q. It is generated by the conjugates of the generators, namely Id, x^{\frac{1}{2}} D_{y}, x^{− \frac{1}{2}} y and

(7.14)  \frac{1}{x} ((x^{2}D_{x} − ν̃)^{2} − ν̃^{2}) + xD_{y}^{2} + x^{−1}y^{2}a(y).

The last two terms in (7.14) are in M̃^{2}. So, following the observation (6.9), we may drop them without changing the spaces I^{(s)}(Õ, M̃). The new generator can then be written

\frac{1}{x} ((x^{2}D_{x} − ν̃)^{2} − ν̃^{2}) = (xD_{x}x − 2ν̃)xD_{x}.

The factor xD_{x}x − 2ν̃ ∈ Ψ_{sc}^{1,0}(X) is elliptic at the base point (0, 0, 0) so can be dropped without changing the test module. We can also add the product of the other two generators, again using (6.9). Thus we see that

(7.15) I^{(s)}(O, M) = e^{iν̃/x} · I^{(s)}(Õ, M′) with

M′ generated by Id, x^{\frac{1}{2}} D_{y}, x^{− \frac{1}{2}} y and 2xD_{x} + yD_{y}.

Now, consider the effect of the parabolic blow up of z, passing from X to X_{z} on

(7.16)  v ∈ x^{s}L_{sc}^{2}(X) with WF_{sc}(v) ⊂ {q̃}, v ∈ I^{(s)}(Õ, M′).

As a neighbourhood of the front face we may take

(7.17)  [0, δ)_{ρ} × \overline{ℝ_{Y}}

where ρ = (x + |y|^{2})^{\frac{1}{2}} = x^{\frac{1}{2}} (1 + |Y |^{2})^{− \frac{1}{2}} , Y = x^{− \frac{1}{2}} y and \overline{ℝ_{Y}} is the radial compactification of the line (so that on \overline{ℝ_{Y}}, 1/|Y | is a defining function for ‘infinity’). Thus we may take ρ_{ff} = ρ and ρ_{o} = (1 + Y^{2})^{−1/2} for explicit boundary defining functions. In terms of coordinates (ρ, Y), the generators of M′ in (7.15) become

(7.18) Id, x^{\frac{1}{2}} D_{y} = D_{Y} + Y (1 + |Y |^{2})^{−1}ρD_{ρ}, x^{− \frac{1}{2}} y = Y, 2xD_{x} + yD_{y} = ρD_{ρ}.

We may replace the second generator by just D_{Y}. Computing the pull-back of the Riemannian measure we see that in region (7.17) we have

(7.19)  Y^{k}D_{Y}^{l} (ρD_{ρ})^{m}β^{∗}v ∈ ρ_{o}^{2s+2}ρ_{ff}^{2s+3/2}L_{b}^{2}([0, δ)_{ρ} × \overline{ℝ_{Y}}) ∀ k, l, m ∈ ℕ_{0}.

Full iterative regularity with respect to Y and D_{Y} in a polynomially weighted L^{2} space is equivalent to regularity in the Schwartz space, and hence to smoothness up to, and rapid vanish at, the old boundary Y = ∞. Thus (7.19) reduces to the statement that β^{∗}v is Schwartz in Y with values in the conormal space in ρ = ρ_{ff} which is precisely the content of (7.9). □

Theorem 7.4. Let q ∈ Min_{+}(λ) be a center for W, i.e. (7.1) holds, and let (x, y) be local coordinates centered at z, with y given by arclength along ∂X. Let X = x^{\frac{1}{2}} and Y = x^{− \frac{1}{2}} y, let h = V′′(z), and let Q be the operator

(7.20)  Q = D_{Y}^{2} − \frac{ν̃}{2}(Y D_{Y} + D_{Y} Y) + \frac{1}{2}hY^{2},

on L^{2}(ff, dY), with eigenvalues β_{j} and normalized eigenfunctions v_{j}, j = 1, 2, . . ..

Suppose that u satisfies (7.2). Then for all N there exist sequences γ_{j}, γ_{j,k}, 1 ≤ k ≤ N which are rapidly decreasing in j (i.e. for all s, there exists C_{s} > 0 such that |γ_{j}|, |γ_{j,k}| ≤ C_{s}j^{−s}, j ≥ 1) and such that

\begin{matrix}u = e^{iν̃/x}v, & v = ∑_{j} X^{1/2}X^{i(β_{j}+V_{1}(0))/ν̃} (γ_{j} + ∑_{k=1}^{N} γ_{j,k}X^{k} + γ′_{j,N}(X)) v_{j}(Y), \\ where & ν̃ = ν(q), V_{1}(y) = \frac{∂V}{∂x} (0, y), γ′_{j,N} ∈ S(ℝ; X^{N}H_{b}^{∞}([0, δ))). \end{matrix} (7.21)

The map from microfunctions u satisfying (7.2) to rapidly decreasing sequences {γ_{j}} is a bijection.

Proof. Combining Propositions 7.1 and 7.3 we have now established the conormal regularity of the outgoing eigenfunctions at a center on the resolved space X_{z}. To deduce the existence of expansions as in (I.20) we consider the lift of the operator P to X_{z}. Using the coordinates X = x^{\frac{1}{2}} and Y = x^{− \frac{1}{2}} y, arising from Riemannian normal coordinates in the boundary, we find that P̃ − λ = e^{−iν̃/x}(P − λ)e^{iν̃/x} takes the form

(7.22) P̃ −λ = e^{−iν̃/x}((x^{2}D_{x})^{2} +ix^{3}D_{x} +x^{2}D_{y}^{2} +x^{2}A(x, y, x^{2}D_{x})+xB(x, y, xD_{y})(xD_{y})

+ xC(x, y)(x^{2}D_{x})(xD_{y}) + V (x, y) − λ)e^{iν̃/x}

= X^{2} (−ν̃(XD_{X} + i/2) + Q + V_{1}(0)) + X^{3}B′(x, y, XD_{X}, D_{Y}),

where Q is as in (7.20), B′ is a differential operator in XD_{X} and D_{Y} of order at most two with coefficients smooth in (x, y), and V_{1}(y) = ∂_{x}V |_{x=0}.

The operator Q is a harmonic oscillator; in fact, conjugation by e^{iν̃Y^{2}/4} gives

(7.23)  e^{iν̃Y^{2}/4}Qe^{−iν̃Y^{2}/4} = D_{Y}^{2} + α^{2}Y^{2}, α = \sqrt{\frac{V′′_{0}(0)}{2} − \frac{ν̃^{2}}{4}} > 0,

where α is real by (7.1). Thus, Q has discrete spectrum with eigenvalues β_{j} = α(2j + 1), and eigenvalues v_{j} ∈ S(ℝ) which we take to be normalized in L^{2}.

The conclusion of (7.9) is that if u is a microlocal eigenfunction then v = e^{−iν̃/x}u has conormal regularity on X_{z} and vanishes rapidly at the old boundary. Even though X = x^{\frac{1}{2}} , and Y = x^{− \frac{1}{2}} y are singular coordinates at the old boundary, the rapid vanishing there means that v(Y, X) is Schwartz in Y with values in a conormal space X^{r}H_{b}^{∞}([0, δ)) in X. The remainder term in (7.22) maps this space into S(ℝ_{Y}; X^{r+3}H_{b}^{∞}([0, δ))). Thus the condition (P − λ)u ∈ Ċ^{∞}(X) becomes the iterative equation

\begin{aligned}(−ν̃(XD_{X} + i/2) + Q + V_{1}(z))v = XB′ & (X^{2}, XY, XD_{X}, D_{Y})v \\ & ∈ S(ℝ_{Y}; X^{r+1}H_{b}^{∞}([0, δ)). \end{aligned}

Since v is Schwartz in Y with values in a conormal space X^{r}H_{b}^{∞}([0, δ)) in X, the eigenfunction expansion

v = ∑_{j} γ_{j}(X)v_{j}(Y)

converges in the space S(ℝ_{Y}; X^{r}H_{b}^{∞}([0, δ))), and the coefficients satisfy

(7.24)  (ν̃XD_{X} + iν̃/2 − β_{j} − V_{1}(z))γ_{j}(X) = f_{j}(X) ∈ X^{r+1}H_{b}^{∞}([0, δ).

It follows that

γ_{j}(X) = X^{1/2}X^{i(β_{j}+V_{1}(z))/ν̃}γ_{j} + γ′_{j}(X)

with γ_{j} constant and γ′_{j} ∈ X^{r+1}H_{b}^{∞}([0, δ)). In fact the sequence f_{j} is rapidly decreasing in j with values in the space in (7.24) so the same is true of the terms γ′_{j} which come from integration. The γ_{j} therefore also form a rapidly decreasing sequence. These arguments can be iterated, giving the asymptotic expansion, meaning that for any N

\begin{matrix}v = ∑_{j} X^{1/2}X^{i(β_{j}+V_{1}(z))/ν̃} (γ_{j} + ∑_{k=1}^{N} γ_{j,k}X^{k} + γ′_{j,N}(X)) v_{j}(Y), \\ γ′_{j,N} ∈ S(ℝ; X^{N}H_{b}^{∞}([0, δ))) \end{matrix} ; (7.25)

where the series in j converges rapidly. If all the leading constants γ_{j} vanish then repeated integration shows that v ∈ Ċ^{∞}(X_{z}) = Ċ^{∞}(X) is rapidly decreasing.

Conversely the γ_{j} in (7.25) form an arbitrary rapidly decreasing sequence since v may be constructed iteratively as in (7.25) and then asymptotic summation, which can be made uniform in the Schwartz parameter j gives a corresponding microlocal eigenfunction.

This completes the proof of the theorem. □

8. Outgoing eigenfunctions at a sink

In this section we consider the structure of microlocally outgoing eigenfunctions at a radial point q ∈ Min_{+}(λ) which is a sink for W, which occurs when

λ > λ_{Hess}(z) = λ_{Hess} = V (z) + 2V′′(z), z = π(q).

In section 1 it was shown that in this case the linearization of W at q has two negative eigenvalues, −2ν̃r_{1} and −2ν̃r_{2}, where 0 < r_{1} < 1/2 < r_{2} < 1, r_{1} + r_{2} = 1 and ν̃ = ν(q). The discussion here closely parallels that in the previous section, so emphasis is placed on the differences. At the end of the section we also discuss the degenerate case, where λ = λ_{Hess} (and r_{1} = r_{2} = 1/2).

As in the previous section, we may assume the microlocal eigenfunctions satisfy (7.2). We again use commutator methods to deduce iterative regularity, now with respect to the Ψ_{sc}^{−∞,0}(X)-module

(8.1) M generated by Id, {x^{−r_{1}}B, B ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(B)(q) = 0},

{x^{−r_{2}}B, B ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(B)|_{L_{2}} = 0} and x^{−1}(P − λ).

Recall here that L_{2} is a smooth curve given by Proposition 1.4.

We separate the resonant (r_{2}/r_{1} = 2, 3, 4, …) and non-resonant cases, although the proof for the resonant case would go through in the non-resonant case as well, since the proof in the latter case is more transparent.

Proposition 8.1. Suppose q is a sink for W, with eigenvalues −2ν̃r_{i}, as above, and suppose that r_{2}/r_{1} is not an integer. Then the module M in (8.1) is a test module in the sense of Definition 6.2, which satisfies (6.14). If u satisfies (7.2) at q, then u ∈ I_{sc}^{(−1/2−ϵ)}(O, M) for all ϵ > 0.

Proof. Choose real symbols a_{j} ∈ C_{c}^{∞}(^{sc}T^{∗}X) such that da_{j}(q) is, restricted to Σ(λ) at q, a non-trivial eigenvector for the linearization of W with eigenvalue −2ν̃r_{j} ∈ ℝ and with a_{2} = 0 on L_{2} near q. Then a_{1}, a_{2} generate, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions vanishing at q. Similarly, a_{2} generates, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions which vanish on L_{2}. Then if A_{j} ∈ Ψ_{sc}^{−∞,−r_{j}}(X) have symbols x^{−r_{j}}a_{j}, j = 1, 2, M is generated by A_{0} = Id, A_{1}, A_{2} and A_{3} = x^{−1}(P − λ) (notice that the operator with symbol x^{−r_{1}}a_{2} is in M, since r_{1} < r_{2}).

By (6.13), the commutator ix^{−1}[A_{j}, P − λ] ∈ Ψ_{sc}^{−∞,−r_{j}}(X) has principal symbol

(8.2)  − ^{sc}H_{p}(x^{−r_{j}}a_{j}) = x^{−r_{j}}e_{j}, j = 1, 2,

where the e_{j} vanish to second order at q (here we exploit a cancellation, similar to that in Remark 7.2). Since ^{sc}H_{p} is tangent to L_{2}, e_{2} in fact vanishes on L_{2}. Thus

\begin{matrix}ix^{−1}[A_{1}, P − λ] = C_{10}x^{1−r_{1}} + C_{11}A_{1} + x^{r_{2}−r_{1}}C_{12}A_{2} + x^{1−r_{1}}C_{13}A_{3}, \\ ix^{−1}[A_{2}, P − λ] = C_{20}x^{1−r_{2}} + C_{22}A_{2} + x^{1−r_{2}}C_{23}A_{3}, \end{matrix} ; (8.3)

with C_{jk} ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(C_{jk})(q) = 0 for all j and all k ≥ 1. Note that [A_{1}, A_{2}] ∈ Ψ_{sc}^{−∞,0}(X) as r_{1} + r_{2} = 1. So M is closed under commutators, and hence is a test module satisfying (6.14).

The statement about u follows exactly as in Proposition 7.1. □

Next we deal with the case when r_{2}/r_{1} = N is an integer, so r_{1}^{−1} = 1/(N +1). Let I be the ideal of C^{∞} functions on C_{∂} vanishing at q. Now consider the Ψ_{sc}^{−∞,0}(X)- module

(8.4) M generated by Id, {x^{−k/(N+1)}B, B ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(B) ∈ I^{k}}, k = 1, …, N

{x^{−N/(N+1)}B, B ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(B)|_{L_{2}} ∈ I^{N}} and x^{−1}(P − λ).

Proposition 8.2. Suppose q is a sink for W, with eigenvalues −2ν̃r_{i}, and suppose that r_{2}/r_{1} = N is an integer. Then the module M in (8.4) is a test module in the sense of Definition 6.2, which satisfies (6.15). If u satisfies (7.2) at q, then u ∈ I_{sc}^{(−1/2−ϵ)}(O, M) for all ϵ > 0.

Proof. Assume first that r_{1} < 1/2 < r_{2}, i.e. that λ > λ_{Hess}. Choose real symbols a_{j} ∈ C_{c}^{∞}(^{sc}T^{∗}X) such that da_{j}(q) is, restricted to Σ(λ) at q, a non-trivial eigenvector for the linearization of W with eigenvalue −2ν̃r_{j} ∈ ℝ, and a_{2} vanishes on L_{2}. Then a_{1}, a_{2} generate, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions vanishing at q. Similarly, a_{2} generates, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions vanishing on L_{2}. Certainly A_{k} ∈ Ψ_{sc}^{−∞,−kr_{1}}(X) with symbol x^{−kr_{1}}a_{1}^{k}, k = 1, 2, …, N, are in the set in (8.4), as is A_{N+1} with symbol x^{−r_{2}}a_{2}, and indeed M is generated by A_{0} = Id, A_{1}, …, A_{N+1} and A_{N+2} = x^{−1}(P − λ).

By (6.13), the commutator ix^{−1}[A_{1}, P − λ] ∈ Ψ_{sc}^{−∞,−r_{1}}(X) has principal symbol

− ^{sc}H_{p}(x^{−r_{1}}a_{1}) = x^{−r_{1}}e_{1},

and ix^{−1}[A_{N+1}, P − λ] ∈ Ψ_{sc}^{−∞,−r_{2}}(X) has principal symbol

− ^{sc}H_{p}(x^{−r_{2}}a_{2}) = x^{−r_{2}}e_{2},

where the e_{j} vanish to second order at q. Since W is tangent to L_{2} to order N, e_{2} = e′_{2} + e′′_{2}, e′_{2} vanishing on L_{2} and e′′_{2} ∈ I^{N}. Thus

(8.5)  \begin{matrix}ix^{−1}[A_{1}, P − λ] = & C_{10}x^{1−r_{1}} + C_{11}A_{1} + x^{r_{2}−r_{1}}C_{1,N+1}A_{N+1} \\ + x^{1−r_{1}}C_{1,N+2}A_{N+2}, \\ ix^{−1}[A_{k}, P − λ] = & C_{k0}x^{1−kr_{1}} + C_{kk}A_{k} + x^{r_{2}−kr_{1}}C_{k,N+1}A_{N+1} \\ + x^{1−kr_{1}}C_{k,N+2}A_{N+2}, k = 2, 3, …, N \\ ix^{−1}[A_{N+1}, P − λ] = & C_{20}x^{1−r_{2}} + C_{N+1,N}A_{N} + C_{N+1,N+1}A_{N+1} \\ + x^{1−r_{2}}C_{N+1,N+2}A_{N+2}, \end{matrix}

with C_{jk} ∈ Ψ_{sc}^{−∞,0}(X),

σ_{∂}(C_{jk})(q) = 0 for all j ≤ k and k ≥ 1,

but σ_{∂}(C_{N+1,N})(q) usually does not vanish. It is not hard to check that M is closed under commutators, and hence is a test module satisfying (6.15).

The statement about u follows exactly as in Proposition 7.1.

If r_{1} = r_{2} = 1/2, there is only one eigenvector, which we may arrange to be da_{1}(q). We choose any a_{2} with da_{2}(q) linearly independent. The argument as above still works, with N = 1. □

Remark 8.3. In the case that r_{2}/r_{1} is an integer, W is not tangent to the curve L_{2}, and it is necessary to add extra generators A_{2}, …, A_{N} to make the module closed under commutators, as the proof above shows. Nevertheless, the enveloping algebra of M in (8.4) is the same as the enveloping algebra of the module in (8.1), so for regularity considerations, we work with (8.1) below, instead of (8.4), even when r_{2}/r_{1} is integral.

We can again interpret this interative regularity more geometrically by introducing a blown up space, although with blow-up of different homogeneities in different variables, as discussed in Section 2. Namely, we consider the blow-up of X at p along the vector field Ṽ = r_{1}^{−1}x∂_{x} + y∂_{y}:

(8.6)  X_{z,r_{1}} = [X; {z}]Ṽ , Ṽ = r_{1}^{−1}x∂_{x} + y∂_{y},

with blow-down map β = β_{r_{1}}. This space is different to that considered in the previous section (unless r_{1} = 1/2), but it ‘looks similar’, and we use similar notation to describe it. It is again a compact manifold with corners, with two boundary curves, the front face ff = β^{−1}(z) and the old boundary . The two boundary curves are the front face ff = β^{−1}(z), created by the blowup, and the closure of the pullback of ∂X \ {z} which we shall refer to as the old boundary. We shall again denote boundary defining functions for these boundary curves by ρ_{ff} and ρ_{o}, and denote the product of boundary defining functions for the other boundary curves of X by ρ′.

We next describe the appropriate conjugation. Since L_{2} is Legendre and has full rank projection to ∂X near q, we can arrange that nearby L_{2} is given by ν = Φ_{2}(y), μ = Φ′_{2}(y), so inside Σ(λ), μ − Φ′_{2}(y) is a defining function for L_{2}.

Proposition 8.4. If q ∈ Min_{+}(λ) is a sink for W in Σ(λ), with M the test module given by (8.1) or (8.4) in a W -balanced neighbourhood of q then for any s, multiplication gives an isomorphism

(8.7) {u ∈ C^{−∞}(X); WF_{sc}(u) ⊂ {q}, u ∈ I^{(s)}(O, M)} ∋ u ⟼

e^{−iΦ_{2}(y)/x}u ∈ (ρ′)^{∞}ρ_{o}^{∞}ρ_{ff}^{(s+1)/r_{1}−1/2}H_{b}^{∞}(X_{z,r_{1}}).

Remark 8.5. Φ_{2} can be replaced by any other smooth function Φ̃_{2} such that Φ_{2} − Φ̃_{2} = O(y^{1/r_{1}}).

Proof. As in the beginning of the proof of Proposition 7.3 we have coordinates in which (7.10) holds. In this case the module M is therefore generated by A_{0} = Id, A_{1} = x^{−r_{1}}y, A_{2} = x^{−r_{2}}(xD_{y} − Φ′_{2}(y)) and A_{3} = x^{−1}(P − λ). Note that Φ_{2}(y) − ν also vanishes on L_{2}, hence x^{−r_{2}}(x^{2}D_{x} + Φ_{2}(y)) ∈ M as well.

Consider the effect of conjugation by e^{iΦ_{2}(y)/x}. This maps each Ψ_{sc}^{m,l}(X) isomorphically to itself and hence

(8.8)  M̃ = e^{−iΦ_{2}(y)/x}Me^{iΦ_{2}(y)/x}

is another test module, but in the open set

(8.9)  Õ = {(ν − Φ_{2}(y), y, μ − Φ′_{2}(y)); (ν, y, μ) ∈ O} ∋ q̃ = (0, 0, 0),

q̃ being the image of q. It is generated by the conjugates of the generators, namely Id, x^{−r_{1}}y, x^{−r_{2}}(xD_{y}) and

(8.10)  x^{−1}P̃ = x^{−1}e^{−iΦ_{2}(y)/x}(P − λ)e^{iΦ_{2}(y)/x}

To compute this generator, we write P − λ as in (7.22) and express (8.10) as (8.11)

x^{−1}((x^{2}D_{x} − Φ_{2})^{2} + ix(x^{2}D_{x} − Φ_{2}) + (xD_{y} + ∂_{y}Φ_{2})^{2} + x^{2}A(x, y, x^{2}D_{x} − Φ_{2})

+xB(x, y, xD_{y} + ∂_{y}Φ_{2})(xD_{y} + ∂_{y}Φ_{2})
+xC(x, y)(x^{2}D_{x} − Φ_{2})(xD_{y} + ∂_{y}Φ_{2}) + V (x, y) − λ)
= x^{−1}(Φ_{2}^{2} + (∂_{y}Φ_{2})^{2} + V_{0}(y) − λ)
−2Φ_{2}(xD_{x}) − iΦ_{2} + 2(∂_{y}Φ_{2})D_{y} − i∂_{y}^{2}Φ_{2} + B(x, y, ∂_{y}Φ_{2})∂_{y}Φ_{2}
−C(x, y)Φ_{2}∂_{y}Φ_{2} + V_{1}(y) + B_{1}y^{2}D_{y} + B_{2}xD_{y}^{2} + B_{3}x^{3}D_{x}^{2}
+B_{4}x^{2}D_{x} + B_{5}xD_{y} + B_{6}y + B_{7}x, B_{j} ∈ Diff_{sc}^{2}(X).

In the nonresonant case, the term proportional to x^{−1} vanishes identically, since Φ_{2} satisfies the eikonal equation Φ^{2} + (∂_{y}Φ)^{2} + V − λ = 0. Eliminating all terms which are in the span of the generators Id, x^{−r_{1}}y, x^{−r_{2}}(xD_{y}) over Ψ_{sc}^{∗,0}(X), as well as (following the observation (6.9)) elements of M^{2}, we are left with the generator

−2Φ_{2}(xD_{x}) + 2(∂_{y}Φ_{2})D_{y}.

A Taylor series analysis of Φ_{2} gives

(8.12)  Φ_{2}(y) = ν̃(1 − r_{1}y^{2}/2 + O(y^{3})).

Thus, 2(∂_{y}Φ_{2})D_{y} is of the form a(y)yD_{y}. This may be written a(y)(x^{−r_{1}}y)(x^{r_{1}}D_{y}), the product of two generators and a smooth function, so we may eliminate this term, again following the observation (6.9). Thus, in the nonresonant case (8.10) may be replaced by the generator xD_{x}. This is true in the resonant case as well, since, although the term proportional to x^{−1} in (8.11) does not vanish, it is O(y^{1/r_{1}}), and x^{−1}y^{1/r_{1}} ∈ M^{1/r_{1}}. In summary, we have shown

(8.13) I^{(s)}(O, M) = e^{iΦ_{2}(y)/x} · I^{(s)}(Õ, M′) with

M′ generated by Id, x^{−r_{1}}y, x^{−r_{2}}(xD_{y}) and xD_{x}.

Now, consider the effect of the inhomogeneous blow up of z, passing from X to X_{z} on

(8.14)  v ∈ x^{s}L_{sc}^{2}(X) with WF_{sc}(v) ⊂ {q̃}, v ∈ I^{(s)}(Õ, M′).

As a neighbourhood of the front face we may take

(8.15)  [0, δ)_{ρ} × \overline{ℝ_{Y}}

where ρ = (x^{2r_{1}} + y^{2})^{1/2}, Y = x^{−r_{1}}y and \overline{ℝ_{Y}} is the radial compactification of the line, where now 1/|Y |^{1/r_{1}} is a defining function for ‘infinity’ on \overline{ℝ_{Y}}. In terms of these coordinates, the generators of M′ in (8.13) become (8.16) Id, x^{−r_{1}}y = Y, x^{r_{1}}D_{y} = D_{Y} + (1 + |Y |^{2})^{−1}Y ρD_{ρ}, r_{1}(1 + Y^{2})^{−1}ρD_{ρ} − r_{1}Y D_{Y} .

A suitable linear combination of the last two generators, namely yD_{y}+r_{1}^{−1}xD_{x} from (8.13), is ρD_{ρ}. Using (6.9) as before to simplify the third generator and computing the pull-back measure we see that in (8.15)

(8.17) Y^{k}D_{Y}^{l} (ρD_{ρ})^{m}β^{∗}v ∈ ρ_{o}^{(s+1)/r_{1}}ρ_{ff}^{(s+1)/r_{1}−1/2}L_{b}^{2}([0, δ)_{ρ} × \overline{ℝ_{Y}}) ∀ k, l, m ∈ ℕ_{0}.

Full iterative regularity with respect to Y and D_{Y} in a polynomially weighted L^{2} space is equivalent to regularity in the Schwartz space, and hence to smoothness up to, and rapid vanish at, the old boundary Y = ∞. Thus (8.17) reduces to the statement that β^{∗}v is Schwartz in Y with values in the conormal space in ρ_{o} which is precisely the content of (8.7). □

We now show that outgoing microlocal eigenfunctions have full asymptotic expansions on X_{z,r_{1}}. For this we recall from [12] that if I ⊂ ℝ is discrete, I ∩ (−∞, a] finite for a ∈ ℝ, and I +ℕ_{0} ⊂ I then A_{phg,ff}^{I}(X_{z,r_{1}}) is the space of polyhomogeneous conormal distributions with index set I, that vanish to infinite order everywhere but on the front face ff. That is, u ∈ A_{phg,ff}^{I}(X_{z,r_{1}}) means there exist ϕ_{t} ∈ C^{∞}(X_{z,r_{1}}), t ∈ I, vanishing to infinite order everywhere but on the front face, such that

u ∼ ∑_{t∈I} ϕ_{t}ρ_{ff}^{t}.

Here the summation is understood as asymptotic summation, i.e. with the notation of (8.17), the difference v = u − ∑_{t∈I, t≤a} ϕ_{t}ρ_{ff}^{t} satisfies estimates

(8.18)  Y^{k}D_{Y}^{l} (ρD_{ρ})^{m}v ∈ ρ_{ff}^{a}L_{b}^{2}([0, δ)_{ρ} × \overline{ℝ_{Y}}) ∀ k, l, m ∈ ℕ_{0}.

Theorem 8.6. Suppose that q ∈ Min_{+}(λ) is a sink of W, and u satisfies (7.2). Let β = r_{2}/2 + iV_{1}(z)/2ν̃. If r_{2}/r_{1} is not an integer, then

(8.19) u_{0} = x^{−β}e^{−iΦ_{2}/x}u ∈ A_{phg,ff}^{I}(X_{z,r_{1}}), I = ℕ_{0} + ℕ_{0} \frac{1}{r_{1}} + ℕ_{0} \frac{2r_{2} − 1}{r_{1}} + ℕ_{0} \frac{r_{2}}{r_{1}} .

In particular, u_{0} vanishes to infinite order off the front face, and is continuous up to the front face. The map sending microfunctions u satisfying (7.2) to u_{0}|_{ff} is a bijection.

If 1/r_{1} is an integer, r_{1} < 1/2, then the same conclusions hold with (8.19) replaced by

(8.20)  u_{0} = x^{−β}x^{icY^{1/r_{1}}} e^{−iΦ_{2}/x}u, Y = y/x^{r_{1}},

where c is a constant determined by the Taylor series, up to order 1/r_{1}, of V and g at z.

Proof. Suppose first that r_{2}/r_{1} is not an integer. Let δ = min(2r_{2} − 1, r_{1}) > 0. We see from (8.11) and (8.12) that

(8.21)  x^{−1}P̃ = −2ν̃(xD_{x} + r_{1}yD_{y} + iβ) + ρ_{ff}^{δ/r_{1}}P_{2},

where P_{2} is a differential operator of degree at most two generated by vector fields tangent to the boundary of X_{z,r_{1}}. By Proposition 8.4, v = e^{−iΦ_{2}/x}u is invariant under such vector fields. Thus, we deduce that

x^{−1}P̃v = −2ν̃(xD_{x} + r_{1}yD_{y} + iβ)v + ρ_{ff}^{δ/r_{1}}P_{2}v ∈ Ċ^{∞}(X),

so with xD_{x}_{|Y} = xD_{x} + r_{1}yD_{y} denoting the derivative keeping Y, rather than y, fixed, we have by Proposition 8.4

(8.22)  (xD_{x}_{|Y} + iβ)v ∈ x^{−1/2+δ′}L_{sc}^{2}(X) ∀ 0 < δ′ < δ.

Writing v = x^{β}ṽ, and noting that Re β = r_{2}/2, this yields

xD_{x}_{|Y} ṽ ∈ x^{−1/2+δ′−r_{2}/2}L_{sc}^{2}(X) = x^{−1/2+δ′−r_{2}/2}L^{2}(\frac{dxdy}{x^{3}} ).

Taking into account the smoothness in Y, and changing the measure, this means for each fixed Y,

∂_{x}ṽ ∈ x^{−1/2+δ′}L^{2}([0, 1)_{x}; dx) ⊂ x^{δ′′}L^{1}([0, 1)_{x}; dx), ∀δ′′ < δ′.

But that implies that ṽ is continuous to X = 0, and after subtracting ṽ(0, Y), the result is bounded by Cx^{δ′′}. This gives that, modulo x^{−1/2+ϵ′}L_{sc}^{2}(X), ϵ′ > 0 small, microlocally near the critical point, u has the form x^{β}e^{iΦ_{2}(y)/x}u_{0}, where u_{0} is smooth on the blown-up space and rapidly vanishing off the front face. A simple asymptotic series construction then yields the asymptotic series described before, and then the uniqueness result Proposition 4.10 shows that u is actually given by such a series.

Now suppose that r_{2}/r_{1} is an integer. We only need minor modifications of the argument given above; we indicate these below. Equation (8.21) must be replaced by

x^{−1}P̃ = −2ν̃(xD_{x} + r_{1}yD_{y} + cx^{−1}y^{1/r_{1}} + iβ) + x^{δ}P_{2},

where here δ = min(2r_{2} − 1, r_{1}) > 0. Equation (8.22) must be replaced by

(8.23)  (xD_{x}_{|Y} + cY^{1/r_{1}} + iβ)v ∈ x^{−1/2+δ′}L_{sc}^{2}(X) ∀0 < δ′ < δ.

Thus, we reach the same conclusion as before if we redefine

ṽ = x^{−β}x^{icY^{1/r_{1}}} v.
□

Remark 8.7. First, we remark that if 1/r_{1} ∉ ℕ, the form of the parameterization (8.19) is unchanged if we replace Φ_{2}, parameterizing the smooth Legendrian L_{2}, by Φ̃_{2} ∈ C^{∞}(X) with Φ_{2} − Φ̃_{2} ∈ I^{N}, N > 1/r_{1}. Indeed, then

(Φ_{2} − Φ̃_{2})/x = (y/x^{r_{1}})^{N}x^{r_{1}N−1}a, a ∈ C^{∞}(X),

so both the form of (8.19) and u_{0}|_{ff} are unaffected by this change of phase functions. In particular, if r_{1} > 1/N, but we let r_{1} → 1/N, the factor x^{r_{1}N−1} decays less and less, and indeed in the limit it takes a different form (8.20).

If now we allow Φ_{2} − Φ̃_{2} ∈ I^{N}, |r_{1}^{−1} − N| < 1 for some N ∈ ℕ, then (8.21) is replaced by

(8.24) x^{−1}e^{−iΦ̃_{2}/x}(P − λ)e^{iΦ̃_{2}/x} = −2ν̃(xD_{x} + r_{1}yD_{y}) + cy^{N}/x + iβ + ρ_{ff}^{δ}P_{2},

where P_{2} is in the enveloping algebra of M̃ and δ > 0. (Note that the y^{N}/x term can be included in P_{2} if N > 1/r_{1}, but not otherwise, and we are interested in taking r_{1} → 1/N!) Proceeding with the argument as in the resonant case, (8.23) is replaced by

(8.25)  (xD_{x}_{|Y} + cx^{r_{1}N−1}Y^{N} + iβ)v ∈ x^{−1/2+δ}L_{sc}^{2}(X).

If r_{1}N − 1 ≠ 1, we can remove the (cx^{r_{1}N−1}Y^{N} + iβ) terms by introducing an integrating factor, namely by writing v = x^{−β}e^{−ic(r_{1}N−1)^{−1}x^{r_{1}N−1}Y^{N}} ṽ, to obtain that ṽ is continuous to x = 0, and the asymptotics take the form

(8.26)  u = x^{β}e^{−ic(r_{1}N−1)^{−1}x^{r_{1}N−1}Y^{N}} e^{iΦ_{2}/x}u_{0}, Y = y/x^{r_{1}},

with u_{0} continuous up to the front face of X_{z,r_{1}}, and (8.20) is the limiting case as r_{1}N → 1. In particular, the special role played by 1/r_{1} ∈ ℕ in the statement of the theorem is partly due to its formulation. Here if N > 1/r_{1}, the factor e^{−ic(r_{1}N−1)^{−1}x^{r_{1}N−1}Y^{N}} tends to 1 in the interior of ff, i.e. does not affect the form of the asymptotics, but if N < 1/r_{1}, it introduces an oscillatory factor with phase −c(r_{1}N − 1)^{−1}y^{N}/x.

Finally, we analyze what happens at the Hessian threshold, i.e. when r_{1} = r_{2} = 1/2. The correct space to describe the asymptotics is not quite the parabolic blowup [X; \{z\}]_{\frac{1}{2}} , on which Y = yx^{−1/2} is smooth, rather the space on which

\frac{y}{x^{1/2} log x} = \frac{Y}{log x}

is smooth. To simplify the statement we only consider the top order of the asymptotics.

Theorem 8.8. Suppose that q ∈ Min_{+}(λ), and λ is at the Hessian threshold for q, so that r_{1}(q) = r_{2}(q) = \frac{1}{2}. Let β as in Theorem 8.6, let Φ_{2}(y) = ν̃(1 − y^{2}/4), where ν̃ = ν(q), and let Y = x^{−1/2}y. Suppose that u satisfies (7.2). Then

(8.27)  u_{0} = x^{−β}(log x)^{1/2}e^{−iΦ_{2}/x}e^{i \frac{ν̃}{4} \frac{Y ^{2}}{log x}} u

is such that

(8.28)  u_{0} − g(Y/ log x) ∈ (log x)^{−1}L^{∞}(X) for some g ∈ S(ℝ).

The map from microfunctions u satisfying (7.2) to g is a bijection.

The full asymptotic expansion, which is not stated above, is in terms of powers of log x and it arises from a stationary phase argument as can be seen from the proof given below. One could instead give a description of u as an oscillatory integral, analogous to how u is described as an oscillatory sum in Theorem 7.4, with an error term in x^{\frac{1}{2} −δ}L^{∞} for all δ > 0. In particular, not only are u, hence all terms of the (logarithmic) asymptotic expansion, determined by g, but the terms only depend on the principal symbol of P − λ near q, modulo I^{2}, and the subprincipal symbol at q.

Proof. We use the computations of Theorem 7.4. Thus, with P̃ = e^{−iν̃/x}P e^{iν̃/x}, and X = x^{1/2},

(8.29) P̃ = X^{2} (−ν̃(XD_{X} + i/2) + Q + V_{1}(z)) + X^{3}B′(X, XD_{X}, Y, D_{Y}),

with Q = e^{iν̃Y^{2}/4}D_{Y}^{2} e^{−iν̃Y^{2}/4},

since α in (7.23) is zero at the Hessian threshold. Thus

(8.30) P̃′ = e^{−iΦ_{2}/x}(P − λ)e^{iΦ_{2}/x}

= X^{2} (−ν̃(XD_{X} + i/2) + D_{Y}^{2} + V_{1}(z)) + X^{3}B′′(X, XD_{X}, Y, D_{Y}).

We thus deduce that

X^{−3} (P̃′ − X^{2}(−ν̃XD_{X} + D_{Y}^{2} − 2iν̃β))

is in the enveloping algebra of M̃. Thus, with u = e^{iΦ_{2}/x}X^{β}v,

(−ν̃XD_{X} + D_{Y}^{2} )v ∈ x^{1−δ}L_{sc}^{2}(X), ∀δ > 0.

The operator on the left hand side is the Schrödinger operator −D_{t} + D_{Y}^{2} after a logarithmic change of variables t = \frac{1}{ν̃} log X. This can now be solved explicitly by taking the Fourier transform in Y, using the fact that v is Schwartz in Y for X bounded away from 0. The solution thus has the form of the inverse Fourier transform of X^{iη^{2}/ν̃}g̃(η), plus faster vanishing terms, where g̃ is Schwartz, and η is the dual variable of Y. This gives an expression of the form (8.27). □

9. Outgoing eigenfunctions at a saddle

We next analyze microlocal eigenfunctions u ∈ E_{mic,+}(q, λ) when q ∈ Max_{+}(λ), hence q is an outgoing saddle point for W. Recall from section 1 that in this case there are always two smooth Legendre curves through q which are tangent to W, L_{1} and L_{2}, and there is a local coordinate v_{i} on L_{i} with v_{i} = 0 at q such that W takes the form −2ν̃r_{1}v_{2}∂_{v_{2}} on L_{2} and −2ν̃r_{2}v_{1}∂_{v_{1}} on L_{1}. Here ν̃ = ν(q) is positive, and r_{i} satisfy r_{1} < 0 and r_{2} > 1. Hence L_{2} ⊂ Φ_{+}({q}) is the outgoing Legendrian in this case, and by microlocalizing u ∈ E_{mic,+}(q, λ) to a W -balanced neighbourhood O of q, we may assume that

(9.1)  u ∈ C^{−∞}(X), O ∩ WF_{sc}((P − λ)u) = ∅, WF_{sc}(u) ∩ O ⊂ L_{2}.

As before we first use commutator methods to deduce iterative regularity, now with respect to the Ψ_{sc}^{−∞,0}(X)-module

(9.2)  M generated by Id, {x^{−1}B, B ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(B)|_{L_{2}} = 0}.

In particular, x^{−1}(P − λ) ∈ M, since σ_{∂}(P − λ) vanishes on L_{2}.

Proposition 9.1. The module M in (9.2) is a test module in the sense of Definition 6.2, which satisfies (6.14). If u satisfies (9.1) at q, then u ∈ I_{sc}^{(−1/2−ϵ)}(O, M) for all ϵ > 0. That is, u is a Legendre distribution microlocally near q.

Proof. Choose real symbols a_{j} ∈ C_{c}^{∞}(^{sc}T^{∗}X) such that a_{j} vanishes on L_{j} in a neighbourhood of q, and such that da_{j}(q) is, restricted to Σ(λ) at q, a non-trivial eigenvector for the linearization of W with eigenvalue −2ν̃r_{j} ∈ ℝ. Then a_{1}, a_{2} generate, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions vanishing at q. Similarly, a_{i} generates, over C_{c}^{∞}(Σ(λ)) and locally near q, the ideal of functions also with differentials vanishing on L_{i}. Certainly A_{1} ∈ x^{−1}Ψ_{sc}^{−∞,0}(X) with symbol x^{−1}a_{2}, is in the set in (9.2) and indeed M is generated by A_{0} = Id, A_{1} and A_{2} = x^{−1}(P − λ).

By (6.13), the commutator i[A_{1}, P − λ] ∈ Ψ_{sc}^{−∞,−1}(X) has principal symbol

− ^{sc}H_{p}(x^{−1}a_{2}) = x^{−1}(2ν̃(r_{2} − 1)a_{2} + e_{2}).

where e_{2} vanishes to second order at q. Since ^{sc}H_{p} is tangent to L_{2}, e_{2} in fact vanishes on L_{2}. Thus

(9.3)  ix^{−1}[A_{1}, P − λ] = C_{10} + C_{11}A_{1} + C_{12}A_{2},

with C_{jk} ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(C_{12})(q) = 0, and σ_{∂}(C_{11})(q) = 2ν̃(r_{2} − 1) > 0. In particular M is closed under commutators, and hence is a test module satisfying (6.14).

To prove the statement about u, note that WF_{sc}^{∗,−1/2−ϵ}(u) ∩ O = ∅ for any W - balanced neighbourhood O of q, by Corollary 5.9. To apply Proposition 6.7, we need to construct a Q satisfying (6.27). We take Q ∈ Ψ_{sc}^{−∞,0}(X) such that

(9.4)  σ_{∂}(Q) = q = χ_{1}(a_{1}^{2})χ_{2}(a_{2})ψ(p),

where χ_{1}, χ_{2}, ψ ∈ C_{c}^{∞}(ℝ), χ_{1}, χ_{2} ≥ 0 are supported near 0, ψ supported near σ, χ_{1}, χ_{2} ≡ 1 near 0 and χ′_{1} ≤ 0 in [0, ∞). Note that a_{2} = 0 on L_{2}, so supp d(χ_{2} ∘ a_{2}) ∩ L_{2} = ∅. On the other hand,

(9.5)  ^{sc}H_{p}χ_{1}(a_{1}^{2}) = 2a_{1}( ^{sc}H_{p}a_{1})χ′_{1}(a_{1}^{2}) = −4ν̃a_{1}(r_{1}a_{1} + e_{1})χ′_{1}(a_{1}^{2}),

with e_{1} vanishing quadratically at q. Moreover, on supp χ′_{1}, a_{2} is bounded away from 0, and r_{1} < 0, so r_{1}a_{1} + e_{1} ≠ 0, provided supp χ′_{1} is sufficiently small. Since r_{1} < 0 and χ′_{1} ≤ 0, 4ν̃r_{1}a_{1}^{2}χ′_{1}(a_{1}^{2}) is positive. Hence, choosing χ_{1} such that (−χ_{1}χ′_{1})^{1/2} is C^{∞}, and supp χ_{1} is sufficiently small, we can write

\begin{matrix}σ_{∂}(i[Q^{∗}Q, P − λ]) = − ^{sc}H_{p}q^{2} = 4ν̃b̃^{2} + f̃, \\ b̃ = (a_{1}(r_{1}a_{1} + e_{1})χ′_{1}(a_{1}^{2})χ_{1}(a_{1}^{2}))^{1/2}χ_{2}(a_{2})ψ(p), supp f̃∩ L_{2} = ∅. \end{matrix} ; (9.6)

Now we can apply Proposition 6.9, using the remark that follows it, to finish the proof. □

Let Φ_{2} parameterize L_{2} near q, as in the previous section. Multiplication by e^{−iΦ_{2}(y)/x} maps L_{2} to the zero section ν = 0, μ = 0. The set of Legendre distributions associated to the zero section is exactly that of distibutions conormal to the boundary. The test module corresponding to this class (which is the conjugate of M under e^{−iΦ_{2}/x}), is M̃ = V_{b}(X), generated by xD_{x} and D_{y}, microlocalized in Õ, the image of O under the multiplication. We thus deduce the following corollary.

Corollary 9.2. Suppose that u satisfies (9.1) for q ∈ Max_{+}(λ). Then e^{−iΦ_{2}(y)/x}u ∈ I_{sc}^{(−1/2−ϵ)}(Õ, M̃) for all ϵ > 0, i.e. its microlocalization to Õ is in x^{1/2−ϵ}H_{b}^{∞}(X) for all ϵ > 0.

Remark 9.3. The L^{2} conormal space x^{s}H_{b}^{∞}(X) is contained in the weighted L^{∞} space with the weight shifted by ϵ; that is, v ∈ x^{s}H_{b}^{∞}(X) gives v ∈ x^{s−ϵ}L^{∞}(X) for all ϵ > 0 (by Sobolev embedding). Conversely, if v ∈ x^{m}L^{∞}(X) ∩ x^{s}H_{b}^{∞}(X) then by interpolation, v ∈ x^{m−ϵ}H_{b}^{∞}(X) for all ϵ > 0.

The equation (P −λ)u ∈ Ċ^{∞}(X) for Legendre functions u reduces to a transport equation. We can obtain this transport equation by a very explicit conjugation of P − λ as the following proposition shows.

Proposition 9.4. There exists an operator A of the form

A = e^{iΦ_{2}/x} ∘ F^{∗} ∘ x^{β}e^{if},

where β = r_{2}/2 + iV_{1}(z)/2ν̃, f ∈ C^{∞}(X), and F^{∗} is pullback by a diffeomorphism F on X, such that

(9.7)  A^{−1}(P − λ)A − P̃_{0} ∈ x^{2}M̃^{2}, P̃_{0} = −2ν̃((x^{2}D_{x}) + r_{1}y(xD_{y})).

Proof. By (8.11) and (8.12), we have

e^{−iΦ_{2}/x}(P − λ)e^{iΦ_{2}/x} =

− 2ν̃((x^{2}D_{x}) + r_{1}y(xD_{y}) + ixβ(y)) + yF (y)(x^{2}D_{x}) + y^{2}G(y)(xD_{y}) mod x^{2}M̃^{2},

where β(0) = β. Since r_{1} < 0, the vector field x∂_{x} + r_{1}y∂_{y} is nonresonant. Hence by a change of coordinates x′ = a(y)x, y′ = b(y)y, we may arrange that the vector field becomes

−2ν̃(((x′)^{2}D_{x′}) + r_{1}y(x′D_{y′})),

modulo terms in x^{2}M̃^{2} and subprincipal terms. We therefore deduce that there is a local diffeomorphism F such that

\begin{matrix}(F^{−1})^{∗}e^{−iΦ_{2}/x}(P − λ)e^{iΦ_{2}/x}F^{∗} − P̃_{1} ∈ x^{2}M̃^{2}, \\ P̃_{1} = −2ν̃((x^{2}D_{x}) + r_{1}y(xD_{y}) + ixβ(y)). \end{matrix} ; (9.8)

Now we let β̃(y) = (β(y) − β(0))/y and set f(y) = ∫_{0}^{y} β̃(z) dz. This yields

(9.9)  x^{−β}e^{−if} P̃_{1}e^{if}x^{β} − P̃_{0} ∈ x^{2}M̃^{2},

which completes the proof of the proposition. □

Remark 9.5. We have not conjugated the operator P − λ microlocally to P̃_{0}; we have only conjugated it to the correct form ‘along L_{2}’ – this suffices because of the a priori conormal estimates. Note also that e^{−iΦ_{2}/x}u ∈ x^{s}H_{b}^{∞}(X) is equivalent to A^{−1}u ∈ x^{s+r_{2}/2}H_{b}^{∞}(X).

Note that (yx^{−r_{1}})^{n}, for n ≥ 0 integral, satisfies P̃_{0}(yx^{−r_{1}})^{n} = 0, and (yx^{−r_{1}})^{n} ∈ x^{s}H_{b}^{∞}(X) for all s < −r_{1}n (where s is increasing with n, since r_{1} < 0). These can be modified to obtain microlocal solutions of A^{−1}(P − λ)Av ∈ Ċ^{∞}(X).

Proposition 9.6. For each integer n ≥ 0, there exists v_{n} ∈ ∩_{s<−r_{1}n}x^{s}H_{b}^{∞}(X) such that v_{n} − (yx^{−r_{1}})^{n} ∈ x^{s′}H_{b}^{∞}(X) for some s′ > −r_{1}n, and such that q ∉ WF_{sc}(A^{−1}(P − λ)Av_{n}).

Proof. Applying A^{−1}(P − λ)A to x^{α}y^{n} yields a function of the form x^{α+2}f′, f′ ∈ C^{∞}(X). The equation P̃_{0}x^{α+1}g = x^{α+2}f′ has a smooth solution g modulo x^{α+3}f′′, f′′ ∈ C^{∞}(X), provided α + 1 is not an integer multiple of r_{1} (and then only a logarithmic factor in x is needed). Applying this argument with α = −r_{1}n, etc., we deduce that the error terms arising from (yx^{−r_{1}})^{n} can be solved away iteratively. □

This result shows that A^{−1}(P − λ)A, hence P − λ, have a series of approximate generalized eigenfunctions v_{n}, resp. u_{n} = Av_{n}, where u_{n} has the form u_{n} = e^{iΦ_{2}/x}x^{β−r_{i}n}ṽ_{n}, with ṽ_{n} polyhomogeneous conormal and continuous up to the boundary.

We proceed to show that for any u satisfying (9.1) there exist constants a_{n} such that u ∼ ∑_{n=0}^{∞} a_{n}u_{n}. The main technical result is the following lemma.

If U ⊂ X is open, we denote by x^{s}H_{b}^{∞}(U) the space of functions consisting of the restrictions of x^{s}H_{b}^{∞}(X) to U.

Lemma 9.7. Suppose that U_{0} is a neighbourhood of z in X. There exists a neighbourhood U ⊂ U_{0} of z such that the following hold.

Suppose that v|_{U} ∈ x^{r}H_{b}^{∞}(U) and x^{−1}P̃_{0}v|_{U} ∈ x^{s}H_{b}^{∞}(U), s > r. Then

(1) If s ≤ 0 then v|_{U} ∈ x^{s−ϵ}H_{b}^{∞}(U) for all ϵ > 0.

(2) If s > 0, then there exists a constant a_{0} such that v|_{U} − a_{0} ∈ x^{r′}H_{b}^{∞}(U) for

any r′ < min(−r_{1}, s).
(3) If r > 0, then v|_{U} ∈ x^{r′}H_{b}^{∞}(U) for all r′ < min(−r_{1}, s).

(4) More generally, if n ≥ 0 is an integer, r > −(n − 1)r_{1}, s > −nr_{1}, then there exists a constant a_{n} such that v|_{U} − a_{n}(x^{−r_{1}}y)^{n} ∈ x^{r′}H_{b}^{∞}(U) for any

r′ < min(−(n + 1)r_{1}, s).

(5) If s > r, r > (n − 1)r_{1}, n ≥ 0 an integer, then v|_{U} ∈ x^{r′}H_{b}^{∞}(U) for all

r′ < min(−nr_{1}, s).

Proof. Let I be a small open interval, 0 ∈ I, and let x_{0} > 0 be small, so that U = [0, x_{0}]_{x} × I_{y} ⊂ U_{0} is a coordinate neighbourhood of z. Thus, v|_{U} ∈ x^{r}H_{b}^{∞}(U) and x^{−1}P̃_{0}v|_{U} ∈ x^{s}H_{b}^{∞}(U), s > r. The integral curves of the vector field P̃_{0} are given by

x^{−r_{1}}y = constant.

Since −r_{1} > 0, if x is increasing along an integral curve then |y| is decreasing, and vice versa. Hence, the integral curve γ through (x, y) ∈ U satisfies γ(T) ∈ U provided x_{0} ≤ x(γ(T)) ≤ x.

Since P̃_{0} is a vector field, v − v|_{x=x_{0}} is given by the integral of f = x^{−1}P̃_{0}v along integral curves of P̃_{0}. Namely, the solution is given by

v(x, y) = −(2ν̃)^{−1} ∫_{x_{0}}^{x} f(t, (\frac{x}{t})^{−r_{1}} y) \frac{dt}{t} + v(x_{0}, (\frac{x}{x_{0}} )^{−r_{1}} y).

Note that the second term, which solves the homogeneous equation, is certainly a polyhomogeneous function on (x, y) down to x = 0, since it only evaluates v at x = x_{0}. Thus, it has a full asymptotic expansion, corresponding to the Taylor series of v at x = x_{0} around y = 0 of the form ∑_{j=0}^{∞} c_{j}(x^{−r_{1}}y)^{j}.

So suppose that s ≤ 0 first. Then the first term (when restricted to U) is in x^{s−δ}L^{∞} for all δ > 0 since f ∈ x^{s}L^{∞} along the restriction of γ to x ≤ x ∘ γ ≤ x_{0}. Hence, v ∈ x^{s−δ}L^{∞} for all δ > 0, so v ∈ x^{s−ϵ}H_{b}^{∞}(X) for all ϵ > 0.

On the other hand, suppose that f ∈ x^{s}L^{∞} for some s > 0. The first term gives a convergent integral, hence is L^{∞}; indeed, it is continuous to x = 0 with limit ∫_{x_{0}}^{0} f(t, 0) \frac{dt}{t}. Let a_{0} = v(x_{0}, 0) + ∫_{x_{0}}^{0} f(t, 0) \frac{dt}{t}. Then v − a_{0} ∈ x^{min(−r_{1},s−ϵ)}L^{∞}(X) for all ϵ > 0, so v − a_{0} ∈ x^{min(−r_{1},s)−ϵ}H_{b}^{∞}(X) by Remark 9.3. To see this, we write

v − a_{0} = ∫_{x_{0}}^{x}[f(t, 0) − f(t, (\frac{x}{t})^{−r_{1}} y)] \frac{dt}{t} = ∫_{x_{0}}^{x} ∫_{0}^{1} (\frac{x}{t})^{−r_{1}} y \frac{∂f}{∂y} (t, ρ (\frac{x}{t})^{−r_{1}} y) dρ \frac{dt}{t},

and estimate the integrand directly. A similar argument works for arbitrary n. □

Corollary 9.8. Let v_{n}, n ≥ 0 integer, be given by Proposition 9.6. Suppose that U_{0} is a neighbourhood of z. There exists a neighbourhood U ⊂ U_{0} of z with the following properties.

Suppose that v ∈ x^{r}H_{b}^{∞}(U) and x^{−1}A^{−1}(P − λ)Av ∈ Ċ^{∞}(U). Then there exists a constant a_{0} such that v − a_{0}v_{0} ∈ x^{r′}H_{b}^{∞}(X) for any r′ < −r_{1}. More generally, if n ≥ 0 is an integer, r > −(n − 1)r_{1}, then there exists a constant a_{n} such that v − a_{n}v_{n} ∈ x^{r′}H_{b}^{∞}(X) for any r′ < −(n + 1)r_{1}.

Proof. First, by Proposition 9.4, A^{−1}(P − λ)A = P̃_{0} + x^{2}E, E ∈ M̃^{2}.

Let r_{0} = sup{r′; v ∈ x^{r′}H_{b}^{∞}(U)}. Thus, v ∈ x^{r′}H_{b}^{∞}(U) for all r′ < r, hence Ev ∈ x^{r′}H_{b}^{∞}(U). Thus, x^{−1}P̃_{0}v ∈ x^{r′+1}H_{b}^{∞}(U) for all r′ < r_{0}.

We can now apply the previous lemma. Namely, suppose first that r_{0} < 0, so x^{−1}P̃_{0}v ∈ x^{r_{0}+1−ϵ}H_{b}^{∞}(U) for all ϵ > 0. Part (i) of the lemma, applied with s = min(0, r_{0} + 1 − ϵ) ≤ 0, shows that v ∈ x^{s−ϵ}H_{b}^{∞}(U) for all ϵ > 0. Thus, either v ∈ x^{−ϵ}H_{b}^{∞}(U) for all ϵ > 0, but this contradicts r_{0} < 0, or v ∈ x^{r_{0}+1−ϵ}H_{b}^{∞}(U) for all ϵ > 0, which in turn contradicts the definition of r_{0}. Thus, r_{0} ≥ 0.

By part (ii) of the lemma, there exists a constant a_{0} such that v−a_{0} ∈ x^{r′}H_{b}^{∞}(U) for any r′ < min(−r_{1}, 1).

Let v′ = v − a_{0}v_{0}, with v_{0} given by Proposition 9.6. Then v′ ∈ x^{r′}H_{b}^{∞}(U) for some r′ > 0 still and it satisfies x^{−1}A^{−1}(P − λ)Av′ ∈ Ċ^{∞}(U). Now let r′_{0} = sup{r′; v′ ∈ x^{r′}H_{b}^{∞}(U)} > 0. As above, this yields x^{−1}P̃_{0}v′ ∈ x^{r′+1}H_{b}^{∞}(U) for all r′ < r′_{0}. Suppose that r′_{0} < −r_{1}. Then, by the part (iii) of the lemma, applied with s = r′_{0} + 1 − ϵ, ϵ > 0, v′ ∈ x^{r′}H_{b}^{∞}(U) for all r′ < min(r_{1}, r′_{0} + 1 − ϵ), hence for some r′ > r′_{0}, since we assumed r′_{0} < −r_{1}. Therefore v′ = v − a_{0} ∈ x^{r′}H_{b}^{∞}(U) for any r′ < −r_{1}, finishing the proof of the first part.

The general case, with n arbitrary, is analogous. □

Theorem 9.9. Suppose q ∈ Max_{+}(λ), O is a W -balanced neighbourhood of q and u satisfies (9.1). Let u_{n} = Av_{n} be the local approximate solutions constructed in Proposition 9.6. Then there exist unique constants a_{n} such that for any u′ with u′ ∼ ∑_{n=0}^{∞} a_{n}u_{n}, O ∩ WF_{sc}(u − u′) = ∅. That is,

(9.10)  u ∼ ∑_{n=0}^{∞} a_{n}u_{n}

microlocally near q. The map u ↦ {a_{n}}_{n∈ℕ} is a bijection from microfunctions satisfying (9.1) to complex-valued sequences. Thus, E_{mic,+}(q, λ) is isomorphic to the space of arbitrary complex-valued sequences, i.e. to ℂ[[x]].

Remark 9.10. The theorem implies that microlocally near q, u ∈ x^{r_{2}/2}L^{∞}, hence u ∈ x^{s}L_{sc}^{2}(X) for all s < r_{2}/2 − 1. Since r_{2} > 1, this is an improvement over x^{−1/2−ϵ}L_{sc}^{2}(X), ϵ > 0.

Proof. Let O_{1} ∋ q be any open set with \overline{O_{1}} ⊂ O, and choose some Q ∈ Ψ_{sc}^{−∞,0}(X) such that WF′_{sc}(Id −Q) ∩ \overline{O_{1}} = ∅, WF′_{sc}(Q) ⊂ O. Let ũ = Qu. Then

WF_{sc}(ũ) ⊂ O ∩ WF_{sc}(u) ⊂ O ∩ L_{2},
WF_{sc}((P − λ)ũ) ⊂ WF′_{sc}([P, Q]) ∩ WF_{sc}(u) ⊂ (O \ \overline{O_{1}}) ∩ L_{2},

(9.11)

hence q ∉ WF_{sc}((P − λ)ũ).

By Corollary 9.2 and Proposition 9.4, v = A^{−1}ũ ∈ x^{r}H_{b}^{∞}(U′), for all r < (1 − r_{2})/2 = r_{1}/2 < 0, and by (9.11), A^{−1}(P − λ)ũ is in Ċ^{∞}(U′) for a sufficiently small neighbourhood U′ of the critical point z ∈ ∂X. Hence we can apply Corollary 9.8 to v = A^{−1}ũ. We deduce that there exists a constant a_{0} such that A^{−1}ũ − a_{0}v_{0} ∈ x^{r′}H_{b}^{∞}(X) for all r′ < −r_{1}. Proceeding iteratively now proves (9.10). Surjectivity follows by asymptotically summing the u_{n} given by Proposition 9.6, and injectivity follows from Corollary 9.8. □

10. Incoming eigenfunctions at an outgoing saddle

Let q ∈ Max_{+}(λ) as before, but now consider microlocally incoming functions u at q, i.e. , u ∈ E_{mic,−}(q, λ). We have Φ_{−}({q}) ⊂ L_{1} locally near q, so by microlocalizing we may assume that

(10.1)  u ∈ C^{−∞}(X), O ∩ WF_{sc}((P − λ)u) = ∅, WF_{sc}(u) ∩ O ⊂ L_{1}

in some W -balanced neighbourhood O of q. Thus, in this section L_{1} plays the role that L_{2} played in the previous section. Essentially all the results go through as before. In particular, let Φ_{1}/x parameterize L_{1}, and consider Ψ_{sc}^{−∞,0}(X)-module

(10.2)  M generated by Id, {x^{−1}B, B ∈ Ψ_{sc}^{−∞,0}(X), σ_{∂}(B)|_{L_{1}} = 0}.

This is a test module, which does not satisfy the positivity estimates of (6.14)- (6.15) – unless we take s > −1/2, which is of no interest since we will not have u ∈ x^{s}L_{sc}^{2}(X) for such s.

The test module conjugate to M under e^{−iΦ_{1}/x} is M̃ = V_{b}(X), generated by xD_{x} and D_{y}. We then deduce the following analog of Proposition 9.4.

Proposition 10.1. There exists an operator Ã having the same structure as in Proposition 9.4 such that

(10.3)  Ã^{−1}(P − λ)Ã − P̃_{0} ∈ x^{2}M̃^{2}, P̃_{0} = −2ν̃((x^{2}D_{x}) + r_{2}y(xD_{y})).

Now r_{2} > 1, so the structure of the Legendre solutions of P̃_{0}w = 0 is different from that considered in the previous section. In particular, (yx^{−r_{2}})^{n}, n ≥ 0, integer satisfies P̃_{0}(yx^{−r_{2}})^{n} = 0, and (yx^{−r_{2}})^{n} ∈ x^{s}H_{b}^{∞}(X) for all s < −r_{2}n. Since r_{2} > 0, these become larger at ∂X as n → ∞. These can still be modified to obtain microlocal solutions of Ã^{−1}(P −λ)Ãv ∈ Ċ^{∞}(X), with the proof essentially identical to that presented in the previous section.

Proposition 10.2. For each integer n ≥ 0, there exists w_{n} ∈ ∩_{s<−r_{2}n}x^{s}H_{b}^{∞}(X) such that w_{n} − (yx^{−r_{2}})^{n} ∈ x^{s′}H_{b}^{∞}(X) for some s′ > −r_{2}n, and such that q ∉ WF_{sc}(Ã^{−1}(P − λ)Ãw_{n}).

Note that only finite linear combinations of these are tempered distributions, since every u ∈ C^{−∞}(X) lies in x^{s}H_{b}^{m}(X) for some s, m. In fact, the space of these finite linear combinations is isomorphic to E_{mic,−}(q, λ) as we prove in Section 12.

11. Microlocal Morse decomposition

Equipped with the microlocal eigenspaces E_{mic,±}(q, λ) defined below (I.15), we can decompose R(λ + i0)(Ċ^{∞}(X)) or more precisely RR_{ess,+}^{∞}(λ) defined in (3.9) into microlocal components. Using the identification (3.10) of the space E_{ess}^{∞}(λ) of ‘smooth eigenfunctions’ (see (I.12)) with RR_{ess,+}^{∞}(λ), this decomposes elements of E_{ess}^{∞}(λ) into microlocal building blocks, and in particular proves Theorem I.2.

Let us start with the perfect Morse case, with z_{min} and z_{max} denoting the minimum and maximum of V_{0}. Denote by ι the identification map

(11.1)  ι : E_{mic,+}(q_{min}^{+}, λ) ⟶ RR_{ess,+}^{∞}(λ).

Namely, by Corollary 5.7, u ∈ E_{mic,+}(q_{min}^{+}, λ), may be regarded as a distribution with WF_{sc}(u) ⊂ {q_{min}^{+}} and f = (P − λ)u ∈ Ċ^{∞}(X), determined up to addition of an element of Ċ^{∞}(X). Then u = R(λ+i0)f since both sides are outgoing, and their difference is a generalized eigenfunction of P, so u can indeed be identified with an element of RR_{ess,+}^{∞}(λ). The map

r : RR_{ess,+}^{∞}(λ) ⟶ E_{mic,+}(q_{max}^{+}, λ)

arises by microlocal restriction. That is, we choose Q ∈ Ψ_{sc}^{0,0}(X) with σ(Q) = 1 in a neighbourhood of q_{max}^{+}, and map u ∈ RR_{ess,+}(λ) to Qu ∈ E_{mic,+}(q_{max}^{+}).

Theorem 11.1 (Theorem I.2). Suppose that dim X = 2, ∂X is connected and V_{0} is perfect Morse. For κ < λ < K, ι is an isomorphism. If K < λ then r is surjective and has null space which restricts isomorphically onto E_{mic,+}(q_{min}^{+}, λ) leading to the short exact sequence

0 ⟶ E_{mic,+}(q_{min}^{+}, λ) ⟶ R_{ess,+}^{∞}(λ) ⟶ E_{mic,+}(q_{max}^{+}, λ) ⟶ 0.

Proof. The injectivity of ι is clear.

If λ < K, then any u ∈ RR_{ess,+}^{∞}(λ) has scattering wavefront set in {ν ≥ 0} ∩ Σ(λ), and is invariant under bicharacteristic flow. Thus, we must have WF_{sc}(u) ⊂ {q_{min}^{+}}. If Q is in Ψ_{sc}^{0,0}(X) with σ(Q) = 1 in a neighbourhood of q_{min}^{+}, then Qu ∈ E_{mic,+}(q_{min}^{+}, λ) and ι(Qu) = u, which shows that ι is an isomorphism in this range.

Next suppose that λ > K. Then r ∘ ι = 0 since q_{max}^{+} ∉ WF_{sc}(ι(u)) for u ∈ E_{mic,+}(q_{min}^{+}, λ). Conversely, if v ∈ RR_{ess,+}^{∞}(λ) maps to 0 under r, then by definition ν ≥ ν(q_{max}^{+}) on WF_{sc}(v), and q_{max}^{+} ∉ WF_{sc}(v), hence by Hörmander’s theorem on the propagation of singularities, WF_{sc}(v) ⊂ {q_{min}^{+}}. Thus, v is a representative of an element of E_{mic,+}(q_{min}^{+}, λ), and is indeed in the range of ι. This shows exactness at RR_{ess,+}^{∞}(λ). Finally, surjectivity of r can be seen as follows. Any element of E_{mic,+}(q_{max}^{+}, λ) has a representative u as in Corollary 5.7. In particular, f = (P − λ)u ∈ Ċ^{∞}(X), WF_{sc}(u) ⊂ {ν ≥ ν(q_{max}^{+})}, so u = R(λ + i0)f, showing surjectivity of r. □

The general situation, i.e. when dim X = 2 and V_{0} is Morse, is not substantially different if none of the bicharacteristic curves L_{+} = L_{+}(q), emanating from q ∈ Max_{+}(λ), hits another q′ ∈ Max_{+}(λ). For each q ∈ Min_{+}(λ) there is an identification map

(11.2)  ι_{q} : E_{mic,+}(q, λ) → RR_{ess,+}^{∞}(λ)

defined as described after (11.1). Similarly, for each q ∈ Max_{+}(λ), there is a restriction map

R_{q} : RR_{ess,+}^{∞}(λ) → E_{mic,+}(q)

as described above. Then similar arguments to above yield

Proposition 11.2. Suppose that dim X = 2, V_{0} is Morse and none of the curves L_{+}(q), q ∈ Max_{+}(λ), hit another q′ ∈ Max_{+}(λ). Then

0 ⟶ ⨁_{q∈Min_{+}(λ)}E_{mic,+}(q, λ) ⟶ R_{ess,+}^{∞}(λ) ⟶ ⨁_{q′∈Max_{+}(λ)}E_{mic,+}(q′, λ) ⟶ 0

is a short exact sequence.

Note that in case V_{0} is perfect Morse, we simply recover Theorem I.2.

In complete generality, i.e. for general Morse V_{0}, the relationship between critical points is more complicated. We introduce a partial order on RP_{+}(λ) corresponding to the flow-out under W.

Definition 11.3. If q, q′ ∈ RP_{+}(λ) we say that q ≤ q′ if q′ ∈ Φ_{+}({q}) and q < q′ if q ≤ q′ but q′ ≠ q. A subset Γ ⊂ RP_{+}(λ) is closed under ≤ if for all q ∈ Γ, we have {q′ ∈ RP_{+}(λ) | q ≤ q′} ⊂ Γ. We call the set {q′ ∈ RP_{+}(λ) | q ≤ q′} the string generated by q.

This partial order relation between two radial points in RP_{+}(λ) corresponds to the existence of a sequence q_{j} ∈ RP_{+}(λ), j = 0, …, k, k ≥ 1, with q_{0} = q, q_{k} = q′ and such that for every j = 0, …, k − 1, there is a bicharacteristic γ_{j} with lim_{t→−∞} γ_{j} = q_{j} and lim_{t→+∞} γ_{j} = q_{j+1}.

We can now define a subspace of RR_{ess,+}^{∞}(λ) associated to a ≤-closed subset Γ ⊂ RP_{+}(λ) by setting

(11.3) RR_{ess,+}^{∞}(λ, Γ)

= {u ∈ C^{−∞}(X); (P − λ)u ∈ Ċ^{∞}(X), WF_{sc}(u) ∩ RP_{+}(λ) ⊂ Γ}/Ċ^{∞}(X).

Notice that the assumption that Γ ⊂ RP_{+}(λ) implies that WF_{sc}(u) ⊂ {ν > 0}. The ‘trivial’ case where Γ = RP_{+}(λ) is the space we are ultimately trying to describe:

(11.4)  RR_{ess,+}^{∞}(λ, RP_{+}(λ)) ≡ RR_{ess,+}^{∞}(λ), λ ∉ Cv(V).

Proposition 11.4. Suppose that Γ ⊂ RP_{+}(λ) is ≤-closed and q is a ≤-minimal element of Γ. Then with Γ′ = Γ \ {q}

\xymatrix{0 \ar[r] & RR_{ess,+}^{∞}(λ, Γ′) \ar[r]^{ι} & RR_{ess,+}^{∞}(λ, Γ) \ar[r]^{r_{q}} & E_{mic,+}(λ, q) \ar[r] & 0}

is a short exact sequence.

Proof. The injectivity of ι follows from the definitions. The null space of the microlocal restriction map r_{q}, which can be viewed as restriction to a W -balanced neighbourhood of q, is precisely the subset of RR_{ess,+}^{∞}(λ, Γ) with wave front set disjoint from {q}, and this subset is RR_{ess,+}^{∞}(λ, Γ′). Thus it only remains to check the surjectivity of r_{q}.

We do so first for the strings generated by q ∈ RP_{+}(λ). For q ∈ Min_{+}(λ), the string just consists of q itself and the result follows trivially. So consider the string S(q) generated by q ∈ Max_{+}(λ). By Corollary 5.7 any element of E_{mic,+}(q, λ) has a representative ũ satisfying (P − λ)ũ ∈ Ċ^{∞}(X) with WF_{sc}(ũ) ⊂ Φ_{+}({q}), which immediately gives surjectivity in this case.

For any ≤-closed set Γ and ≤-minimal element q, the string S(q) is contained in Γ, so the surjectivity of r_{q} follows in general. □

Notice that we can always find a sequence ∅ = Γ_{0} ⊂ Γ_{1} ⊂ … ⊂ Γ_{n} = RP_{+}(λ), of ≤-closed sets with Γ_{j} \ Γ_{j−1} consisting of a single point q_{j} which is ≤-minimal in Γ_{j}: we simply order the q_{i} ∈ RP_{+}(λ) so that ν(q_{1}) ≥ ν(q_{2}) ≥ …, and set Γ_{i} = {q_{1}, …, q_{i}}. Then Proposition 11.4 implies the following

Theorem 11.5. Suppose that ∅ = Γ_{0} ⊂ Γ_{1} ⊂ … ⊂ Γ_{n} = RP_{+}(λ), is as described in the previous paragraph. Then

(11.5)  {0} ⟶ R_{ess,+}^{∞}(λ, Γ_{1}) ↪ … ↪ R_{ess,+}^{∞}(λ, Γ_{n−1}) ↪ R_{ess,+}^{∞}(λ),

with

(11.6)  R_{ess,+}^{∞}(λ, Γ_{j})/R_{ess,+}^{∞}(λ, Γ_{j−1}) = E_{ess,+}(q_{j}, λ), j = 1, 2, …, n.

This theorem shows that there are elements of E_{ess,+}^{∞}(λ) associated to each radial point q. The parameterization of the microlocal eigenspaces shows that in some sense, there are ‘more’ eigenfunctions associated to the minima than to the maxima. One way to make this precise is to show that the former class is dense in E_{ess}^{∞}(λ), in the topology of C^{−∞}(X), or indeed in the topology of E_{ess}^{s}(λ) for s > 0 small. We define

(11.7)  E_{Min,+}^{∞}(λ) = {u ∈ E_{ess}^{∞}(λ); WF_{sc}(u) ∩ RP_{+}(λ) ⊂ Min_{+}(λ)} .

Proposition 11.6. Let s_{0} = −1/2 + min{r_{2}(q)/2; q ∈ Max_{+}(λ)} > 0. Then for any λ ∉ Cv(C) and 0 < s < s_{0}, E_{Min,+}^{∞}(λ) is dense in E_{ess}^{∞}(λ) in the topology of E_{ess}^{s}(λ) given by Proposition 3.12.

Proof. We first suppose that

(11.8)  λ is not in the point spectrum of P.

Let u ∈ E_{ess}^{∞}(λ). We know from Theorem 9.9 (see the remark following it) that

WF_{sc}^{∗,s−1/2}(u) ⊂ Min_{−}(λ) ∪ Min_{+}(λ), s < s_{0},

i.e. the wave front set of u relative to x^{s−1/2}H_{sc}^{∞}(X) is localized at the radial points over the minima, provided that s < s_{0}; in particular we can take s > 0.

Now choose Q ∈ Ψ_{sc}^{−∞,0}(X) which has WF′_{sc}(Q) localized near Min_{+}(λ) and WF′_{sc}(Id −Q) disjoint from Min_{+}(λ). Then

WF_{sc}^{∗,s−1/2}(u − Qu) ⊂ Min_{−}(λ), s < s_{0}.

Let P̃ = Δ + Ṽ be the operator constructed in Lemma 5.3 with ν̃ large, so that λ−ν̃^{2} < inf V. Then every radial point q ∈ \begin{matrix} \\ RP & + \end{matrix}(λ) of P̃ lies above a minimum of Ṽ (i.e. \begin{matrix} \\ RP & + \end{matrix}(λ) = \begin{matrix} \\ Min & + \end{matrix}(λ)). Let Π̃ be the orthogonal projection off the L^{2}-nullspace of P̃ − ev, so Id −Π̃ is a finite rank projection to a subspace of Ċ^{∞}(X). Since WF′_{sc}([P, Q]) ⊂ WF′_{sc}(Q) ∩ WF′_{sc}(Id −Q), we have

WF_{sc}^{∗,s−1/2}(u) ∩ WF′_{sc}([P, Q]) = ∅, s < s_{0}.

Moreover,

WF′_{sc}(P − P̃) ∩ WF′_{sc}(Q) = ∅,

and (P − λ)Qu = [P, Q]u + Q(P − λ)u, so

(P − λ)Qu, (P̃ − λ)Qu ∈ x^{s+1/2}H_{sc}^{∞}(X), s < s_{0}.

Here we can take s > 0, hence we can apply R(λ − i0) to these functions, and

WF_{sc}^{∗,s−1/2}(R(λ − i0)(P − λ)Qu) ⊂ Min_{−}(λ).

We claim that

(11.9)  u = v − R(λ − i0)(P − λ)v, v = Qu.

To see this, let ũ = u − (v − R(λ − i0)(P − λ)v). For s < s_{0}, WF_{sc}^{∗,s−1/2}(ũ) lies in Min_{−}(λ), since this is true both for u − v and for R(λ − i0)(P − λ)v. Moreover, (P − λ)ũ = 0, so the uniqueness theorem, Theorem 3.2, shows that ũ = 0.

Since v = Qu has wavefront set confined to {ν > 0} and hence is outgoing,

Π̃v = R̃(λ + i0)(P̃ − λ)v.

Thus,

u = (Id −Π̃)Qu + R̃(λ + i0)w − R(λ − i0)(P − λ)R̃(λ + i0)w,
where w = (P̃ − λ)Qu.

Now consider u′_{j} = ϕ(x/r_{j})u, where ϕ ∈ C^{∞}(ℝ) is identically 1 on [2, +∞), 0 on [0, 1], and r_{j} > 0 satisfy lim_{j→∞} r_{j} = 0. Thus, u′_{j} ∈ Ċ^{∞}(X), and

(11.10)  \begin{matrix}u_{j} = (Id −Π̃)Qu′_{j} + R̃(λ + i0)w_{j} − R(λ − i0)(P − λ)R̃(λ + i0)w_{j}, \\ w_{j} = (P̃ − λ)Qu′_{j} = (P̃ − P)Qu′_{j} + [P, Qϕ(x/r_{j})]u \end{matrix}

satisfy (P − λ)u_{j} = 0 and WF_{sc}(u_{j}) ∩ {ν ≥ 0} ⊂ Min_{+}(λ), so u_{j} ∈ E_{Min,+}^{∞}(λ) ⊂ E_{ess}^{∞}(λ). Since w_{j} → w and (P − λ)Qu′_{j} → (P − λ)Qu in x^{s+1/2}L_{sc}^{2}(X), we have u_{j} → u in x^{r}L_{sc}^{2}(X), for any r < −1/2 by Theorem 3.3. Moreover, if B is as in (3.13) then since WF′_{sc}(B) ∩ RP(λ) = ∅ and WF_{sc}(w) ∩ RP(λ) = ∅, we have Bu_{j} → Bu in x^{s−1/2}L_{sc}^{2}(X) by Theorem 3.1, proving convergence in E_{ess}^{s}(λ).

We now indicate the modifications necessary if (11.8) is not satisfied. Let Π denote the orthogonal projection off the L^{2} λ-eigenspace. Equation (11.9) must be replaced by u = Πv − R(λ − i0)(P − λ)v. Then if we define u_{j} by

u_{j} = ΠR̃(λ + i0)w_{j} − R(λ − i0)(P − λ)R̃(λ + i0)w_{j},

instead of (11.10), the argument goes through. □

12. Pairing and duality

We define a basic version of ‘Green’s pairing’ in this context on

(12.1)  Ẽ^{−∞}(λ) = {u ∈ C^{−∞}(X); (P − λ)u ∈ Ċ^{∞}(X)} .

Namely

(12.2)  B̃(u_{1}, u_{2}) = i (⟨(P − λ)u_{1}, u_{2}⟩ − ⟨u_{1}, (P − λ)u_{2}⟩) , u_{i} ∈ Ẽ^{−∞}(λ).

The imaginary factor is inserted only to make the pairing sesquilinear,

\overline{B̃(u_{1}, u_{2})} = B̃(u_{2}, u_{1}).

Observe that

B̃(u, v) = 0 if u ∈ Ċ^{∞}(X)

since then integration by parts is permitted.

More significantly we define a sesquilinear form on E_{ess}^{∞}(λ) using B̃. Assuming that λ ∉ Cv(V), choose

(12.3) A ∈ Ψ_{sc}^{0,0}(X) such that WF′_{sc}(A) ∩ Σ(λ) ⊂ {ν > −a(λ)} and

WF′_{sc}(Id −A) ∩ Σ(λ) ⊂ {ν < a(λ)} = ∅

where a(λ) is as in Remark 1.11. If u ∈ E_{ess}^{∞}(λ) then WF_{sc}((P − λ)Au) = ∅ so Au ∈ Ẽ^{−∞}(λ). This allows us to define

(12.4)  B : E_{ess}^{∞}(λ) × E_{ess}^{∞}(λ) ⟶ ℂ, B(u_{1}, u_{2}) = B̃(Au_{1}, Au_{2}).

Lemma 12.1. If λ ∉ Cv(V) and A satisfies (12.3) then B in (12.4) is independent of the choice of A and defines a non-degenerate sesquilinear form.

Proof. The space of cut-off operators satisfying (12.3) is convex. If we consider the linear homotopy between two of them and denote the bilinear form B_{t} for t ∈ [0, 1] then

\frac{d}{dt}B_{t}(u_{1}, u_{2}) = B̃(\frac{d}{dt}A_{t}u_{1}, A_{t}u_{2}) + B̃(A_{t}u_{1}, \frac{d}{dt}A_{t}u_{2}) = 0

since \frac{d}{dt}A_{t}u_{i} ∈ Ċ^{∞}(X) for i = 1, 2.

The non-degeneracy of B follows by rewriting it

(12.5)  B(u_{1}, u_{2}) = i⟨(P − λ)Au_{1}, u_{2}⟩, u_{1}, u_{2} ∈ E_{ess}^{∞}(λ).

Indeed the difference between the right side of (12.5) and (12.4) is

i (⟨(P − λ)Au_{1}, (Id −A)u_{2}⟩ + ⟨Au_{1}, (P − λ)Au_{2}⟩) .

In the first term the scattering wavefront sets of Au_{1} and (Id −A)u_{2} do not meet, so integration by parts is permitted and, u_{2} being an eigenfunction, this difference vanishes. If (12.5) vanishes for all u_{1} it follows that ⟨f, u_{2}⟩ = 0 for all f ∈ Ċ^{∞}(X) since it vanishes for f ∈ (P −λ)Ċ^{∞}(X)⊕E_{pp}(λ) which, by Propositions 3.7 and 3.10 is a complement to (P −λ)AE_{ess}^{∞}(λ) in Ċ^{∞}(X). This proves the non-degeneracy. □

Lemma 12.2. The pairing B extends to a non-degenerate pairing and topological duality

(12.6)  B : E_{ess}^{∞}(λ) × E_{ess}^{−∞}(λ) ⟶ ℂ, B(u, v) = B̃(Au, Av).

Moreover, for all s ∈ ℝ, B extends to a nondegenerate pairing

(12.7)  B : E_{ess}^{s}(λ) × E_{ess}^{−s}(λ) ⟶ ℂ, B(u, v) = B̃(Au, Av).

Proof. The extension of the pairing to (12.6) follows directly from (12.5), since Au_{1} ∈ Ċ^{∞}(X) when u_{1} ∈ E_{ess}^{∞}(λ). The same argument shows that it is nondegenerate. To see that it is a topological pairing observe that the Fréchet topology on E_{ess}^{∞}(λ) arises from its identification with the quotient of Ċ^{∞}(X) by the closed subspace (Δ + V − λ)Ċ^{∞}(X). Thus an element of the dual space may be identified with a distribution u ∈ C^{−∞}(X) which vanishes on (Δ + V − λ)Ċ^{∞}(X), i.e. is an element of E_{ess}^{−∞}(λ). The pairing between E_{ess}^{−∞}(λ) and E_{ess}^{∞}(λ) defined in this way is precisely B.

A similar argument applies to the spaces of eigenfunctions of finite regularity. The square of the Hilbert norm on E_{ess}^{s}(λ) may be taken to be

(12.8)  ∥Au∥_{x^{s−1/2}L^{2}}^{2} + ∥u∥_{x^{−N} L^{2}}^{2}

where A is as in (12.3) and N is sufficiently large, in particular N > max\{−s+ \frac{1}{2}, \frac{1}{2}\}. Writing the pairing in the form

(12.9)  B(u_{1}, u_{2}) = i⟨[P, A]u_{1}, u_{2}⟩

shows its continuity with respect to (12.8) (for a different A which is microlocally the identity on the essential support of the operator in (12.9).) The non-degeneracy follows much as before. □

As well as these global forms of the pairing, we can define versions of it on the components of the microlocal Morse decomposition. Namely, for q ∈ RP_{+}(λ), choose Q ∈ Ψ_{sc}^{0,0}(X) with q ∉ WF′_{sc}(Id −Q) and WF′_{sc}(Q) ∩ Σ(λ) supported in a W -balanced neighbourhood O of q. Then consider

(12.10)  Ẽ_{mic,+}(q, λ) × Ẽ_{mic,−}(q, λ) ∋ (u, v) ⟼ B̃(Qu, Qv).

The pairings on the right are well defined. Indeed, for such (u, v),

(12.11)  WF_{sc}(u) ∩ WF_{sc}(v) ∩ O ⊂ {q},

since WF_{sc}(u) ∩ O ⊂ Φ_{+}({q}) and WF_{sc}(v) ∩ O ⊂ Φ_{−}({q}). Moreover, q ∉ WF_{sc} ((P − λ)Qu) and q ∉ WF_{sc} ((P − λ)Qv) , so

(12.12) WF_{sc} ((P − λ)Qu) ∩ WF_{sc}(Qv) = WF_{sc} ((P − λ)Qv) ∩ WF_{sc}(Qu) = ∅,

so B in (12.6) is well-defined.

If Q′ is any other microlocal cutoff with the same properties then q ∉ WF′_{sc}(Q − Q′), hence WF_{sc}((Q − Q′)u) ∩ WF_{sc}(Qv) = ∅, so B̃(Qu − Q′u, Qv) = 0. Similarly, B̃(Qu, Qv − Q′v) = 0, so B̃(Qu, Qv) = B̃(Q′u, Q′v).

If we define a new pairing B′ as in (12.6), but using the operator P̃ from Lemma 5.3 in place of P, then B′(Qu, Qv) = B̃(Qu, Qv) provided that O ∩ WF′_{sc}(P − P̃) = ∅.

Proposition 12.3. For each q ∈ RP_{+}(λ) the sesquilinear form (12.10) descends to a non-degenerate pairing

(12.13)  B̃_{q} : E_{mic,+}(q, λ) × E_{mic,−}(q, λ) ⟶ ℂ.

Proof. The terms involved in the quotients all have wavefront set disjoint from WF′_{sc}(Q), so integration by parts shows that (12.13) is independent of choices.

To show non-degeneracy, suppose that q is a minimal element of RP_{+}(λ) with respect to the partial order ≤ of Definition 11.3. Then every element v of Ẽ_{mic,−}(q, λ) has a representative in E_{ess}^{−∞}(λ) with WF_{sc}(v) ⊂ {ν ≤ ν(q)} and WF_{sc}(v) ∩ {ν = ν(q)} = {q}. By Lemma 12.2, there exists u ∈ E_{ess}^{∞}(λ) such that B̃(u, v) ≠ 0. Since q is ≤-minimal, we have WF_{sc}(u) ∩ WF_{sc}(v) ⊂ {q}. Hence, B̃(u, v) = B̃(Qu, Qv). Thus, Qu is an element of E_{mic,+}(q, λ) with B̃(Qu, v) ≠ 0.

If q is not ≤-minimal, then we use Lemma 5.3 with ν̃ smaller than ν(q), but larger than ν(q′) for any q′ ∈ RP_{+}(λ) with ν(q′) smaller than ν(q). Then q ∈ { \atop RP_{+}(λ)}, and Ṽ = V in a neighbourhood of π(q), so E_{mic,±}(q, λ) is the same space irrespective of whether we consider P or P̃. However, q is ≤-minimal for P̃, so the result follows from the previous paragraph. □

Proposition 12.4. Suppose that q ∈ Max_{+}(λ) and O is a W -balanced neighbourhood of q in ^{sc}T_{Y}^{∗} X. Let u_{n} = Av_{n} ∈ E_{mic,+}(q, λ) and ũ_{n} = Ãw_{n} ∈ E_{mic,+}(q, λ) be the local approximate outgoing, resp. incoming, solutions constructed in Proposition 9.6, resp. Proposition 10.2. Then

\begin{matrix}B̃_{q}(u_{n}, ũ_{m}) = 0 if n > m, \\ B̃_{q}(u_{n}, ũ_{n}) ≠ 0 ∀ n. \end{matrix}

Proof. This follows from a straightforward calculation, using the asymptotic expansions, similarly to the proof of Proposition 12.6 below. □

We can thus renormalize ũ_{m} inductively by letting

ũ′_{m} = \frac{1}{B̃_{q}(u_{m}, ũ_{m})} (ũ_{m} − ∑_{n<m} B̃_{q}(u_{n}, ũ_{m}) ũ′_{n}) .

Then B̃_{q}(u_{n}, ũ′_{m}) = 0 if n ≠ m and B̃_{q}(u_{m}, ũ′_{m}) = 1.

An immediate consequence of non-degeneracy is the following theorem.

Theorem 12.5. Suppose that q ∈ Max_{+}(λ) and O is a W -balanced neighbourhood of q in ^{sc}T_{Y}^{∗} X. If u ∈ E_{mic,−}(q, λ) is a microlocally incoming eigenfunction at q, then there exist unique constants a_{n} such that a_{n} ≠ 0 for only finitely many n, and

(12.14)  O ∩ WF_{sc}(u − ∑ a_{n}ũ_{n}) = ∅.

Thus, E_{mic,−}(q, λ) is isomorphic to the space of finite complex-valued sequences, ℂ[x].

Proof. Choose s > 0 such that u ∈ W′_{s} = E_{mic,−}(q, λ) ∩ x^{−s−1/2}L^{2}(X). Since W′_{s} automatically annihilates E_{mic,+}(q, λ)∩x^{s−1/2}L^{2}(X), we deduce that B̃_{q}(u_{n}, u) = 0 for n sufficiently large. Now u′ = u − ∑_{n} B̃_{q}(u_{n}, u)ũ′_{n} satisfies B̃_{q}(u_{n}, u′) = 0 for all n. The non-degeneracy of B̃_{q} and the structure theorem for E_{mic,+}(q, λ), Theorem 9.9, thus imply that u′ = 0. Since the ũ′_{n} are finite linear combinations of ũ_{n}, the proof is complete. □

We can compute the form of B explicitly on the subspace E_{Min,+}^{∞}(λ). Recall that for each q ∈ Min_{+}(λ) there is a map

M_{+}(q, λ) : E_{Min,+}^{∞}(λ) ⟶ S(ℝ_{r(q)})

given by (I.21) or (I.23), which is an isomorphism on E_{mic,+}(q).

Proposition 12.6. If λ ∉ Cv(V) then for u_{1}, u_{2} ∈ E_{Min,+}^{∞}(λ), (12.15)

B(u_{1}, u_{2}) = 2 ∑_{q∈Min_{+}(λ)} \sqrt{λ − V (π(q))} ∫_{ℝ_{r(q)}} M_{+}(q, λ)(u_{1})\overline{M_{+}(q, λ)(u_{2})} ω_{q},

where ω_{q} is the density on the front face of the blow-up induced by the Riemannian density of h.

Proof. Let χ ∈ C^{∞}(ℝ) be identically 0 near 0 and identically 1 on [1, +∞), and let χ_{r}(x) = χ(x/r). Then, with A as in (12.3),

B(u_{1}, u_{2}) = i lim_{r→0}(⟨χ_{r}(P − λ)Au_{1}, Au_{2}⟩ − ⟨χ_{r}Au_{1}, (P − λ)Au_{2}⟩)

= −i lim_{r→0}⟨[P, χ_{r}]Au_{1}, Au_{2}⟩ = −i lim_{r→0} ∫_{X}([P, χ_{r}]Au_{1})\overline{Au_{2}} dg,

(12.16)

since for r > 0,

⟨(P − λ)χ_{r}Au_{1}, Au_{2}⟩ = ⟨χ_{r}Au_{1}, (P − λ)Au_{2}⟩.

Since [P, χ_{r}] → 0 strongly as a map H_{sc}^{1,s−1/2}(X) ⟶ H_{sc}^{0,s+1/2}(X), for s ∈ ℝ, it follows that only microlocal regions where at least one of Au_{1} and Au_{2} is not in H_{sc}^{0,−1/2}(X) contribute. That is, we can insert microlocal cut-offs near the minimal radial points and thereby localize the computation to a single radial point.

So, let q ∈ Min_{+}(λ), let z = π(q), and assume that λ is above the Hessian threshold for q. Using the asymptotic form of Au_{1} and Au_{2} given by Theorem 8.6, we need to compute

−i lim_{r→0} ∫ [Δ, χ_{r}](e^{iΦ_{2}/x}x^{β}v_{1})e^{−iΦ_{2}/x}x\overline{β}\overline{v_{2}} \frac{dx}{x^{2}} \frac{ω}{x^{r_{2}}} ,

where ω is the density on the front face of the blow-up induced by the Riemannian density of h. Thus, ω = |dh(z)|/x^{r_{1}}. Here, |dx| is fixed at the boundary by the requirement that the scattering metric takes the form (I.1), so this is a well-defined density on the front face of X_{z,r_{1}}. In the computation the commutator [Δ, χ_{r}] may be replaced by 2(x^{2}D_{x}χ_{r})(x^{2}D_{x}) modulo terms that vanish in the limit, and v_{1}, v_{2} may be restricted to the front face. In addition, x^{2}D_{x} must fall on e^{iΦ_{2}/x}, or the limit vanishes. Finally, Re β = r_{2}/2. Hence, the limit is

2iΦ_{2}(z)(∫ v_{1}|_{ff} \overline{v_{2}|_{ff}} ω) (lim_{r→0} ∫_{0}^{∞}(D_{x}χ_{r}) dx) = 2\sqrt{λ − V (z)}(∫ v_{1}|_{ff} \overline{v_{2}|_{ff}} ω).

Since v_{j}|_{ff} = M_{+}(q, λ)(u_{j}), this gives the stated result if λ is above the Hessian threshold of z.

Now suppose that the λ is below the Hessian threshold of q. Then the asymptotics of Au_{i} are given by Theorem 7.4 and take the form

e^{iν̃/x} ∑_{j} X^{1/2}X^{i((2j+1)α+γ)}γ_{j}v_{j}(Y), ν̃ = ν(z) = \sqrt{λ − V (z)},

modulo terms whose contribution vanishes in the limit r → 0, and where the v_{j}, j = 1, 2, … are orthonormal eigenfunctions of a harmonic oscillator. Here γ_{j} = γ_{1,j} (for Au_{1}) or γ_{2,j} (for Au_{2}) is a Schwartz sequence in j, hence interchanging the order of various integrals and sums is permitted. Again, the commutator [Δ, χ_{r}] may be replaced by 2(x^{2}D_{x}χ_{r})(x^{2}D_{x}) and x^{2}D_{x} must fall on e^{iν̃/x}, to yield terms that do not vanish as r → 0. Now we interchange the integral in Y and the summations, and use the fact that the v_{k} are orthonormal, to conclude that the limit is

2\sqrt{λ − V (z)} ∑_{k=1}^{∞} γ_{1,k} \overline{γ_{2,k}}.

Since this is also equal to

2\sqrt{λ − V (z)} ∫ M_{+}(q, λ)(u_{1}) \overline{M_{+}(q, λ)(u_{2})}, M_{+}(q, λ)(u_{i}) = ∑_{j=1}^{∞} γ_{i,j}v_{j},

the proof is complete. □

Corollary 12.7. For λ ∉ Cv(V), the maps M_{+}(q, λ) extend continously to

(12.17)  M_{+}(q, λ) : E_{ess}^{0}(λ) ⟶ L^{2}(ℝ_{r(q)}).

The pairing B is positive definite, hence it defines a norm on E_{ess}^{0}(λ) satisfying (12.18)

0 ≤ ∥u∥_{0}^{2} = 2 ∑_{q∈Min_{+}(λ)}\sqrt{λ − V (π(q))}∥M_{+}(q, λ)u∥^{2} = B(u, u) = i⟨[P − λ, A]u, u⟩

for any A ∈ Ψ_{sc}^{−∞,0}(X) as in (12.3).

Proof. Equation (12.18) holds for u ∈ E_{Min,+}^{∞}(λ), with the positivity coming from Proposition 12.6. Since B extends to a pairing on E_{ess}^{0}(λ), which is continuous in the topology T_{3}^{0}, (12.17) follows from the density statements of Proposition 11.6 and Corollary 3.13 (in T_{3}^{0} in the latter case). As all other expressions extend to continuous bilinear maps on E_{ess}^{0}(λ), the density statements of Proposition 11.6 and Corollary 3.13 show that (12.18) remains true. Since B is non-degenerate and it is a semi-norm, it follows that B is in fact positive definite, hence a norm. □

Lemma 12.8. Suppose that λ ∉ Cv(V). For r < −1/2 there exists C_{r} > 0 such that for u ∈ E_{ess}^{0}(λ), ∥u∥_{x^{r}L_{sc}^{2}(X)}^{2} ≤ C_{r}B(u, u).

Proof. For simplicity of notation, suppose first that λ ∉ σ_{pp}(P).

For f ∈ x^{s}L_{sc}^{2}(X), s > 1/2, let

ũ = R(λ + i0)f − R(λ − i0)f.

For u ∈ E_{ess}^{0}(λ) then

i⟨u, f⟩ = B̃(u, R(λ + i0)f) = B̃(Au, R(λ + i0)f) = B̃(Au, ũ) = B(u, ũ).

By Cauchy-Schwarz and Corollary 12.7,

B(u, ũ) ≤ B(u, u)^{1/2}B(ũ, ũ)^{1/2}.

But

B(ũ, ũ) = B̃(R(λ + i0)f, R(λ + i0)f) = i(⟨f, R(λ + i0)f⟩ − ⟨R(λ + i0)f, f⟩),

so by the limiting absorption principle,

B(ũ, ũ) ≤ C̃_{s}∥f∥_{x^{s}L^{2}}^{2}.

Combining these results gives that

(12.19)  |⟨u, f⟩| ≤ C̃_{s}^{1/2}B(u, u)^{1/2}∥f∥_{x^{s}L^{2}}.

Now for r < −1/2, u ∈ E_{ess}^{0}(λ), let f = x^{−2r}u ∈ x^{s}L_{sc}^{2}(X) for all s < −2r − 1/2, in particular for s = −r > 1/2. Applying (12.19) yields

∥u∥_{x^{r}L^{2}}^{2} ≤ C̃_{−r}^{1/2}B(u, u)^{1/2}∥x^{−2r}u∥_{x^{−r}L^{2}} = C̃_{−r}^{1/2}B(u, u)^{1/2}∥u∥_{x^{r}L^{2}}.

Cancelling a factor of ∥u∥_{x^{r}L^{2}} from both sides proves the lemma if λ ∉ σ_{pp}(P).

If λ ∈ σ_{pp}(P), the calculations up to and including (12.19) work with f replaced by Πf, Π the orthogonal projection off E_{pp}(λ). Take f = x^{−2r}u again and use the identity Πu = u to finish the proof in this case. □

An immediate consequence is the following proposition.

Proposition 12.9. Suppose that λ ∉ Cv(V). The norm (12.18) on E_{ess}^{0}(λ) is equivalent to the the norm

∥u∥_{T_{3}^{0}} = ∥u∥_{3} = (∥u∥_{x^{r}L_{sc}^{2}(X)}^{2} + ∥B′u∥_{x^{−1/2}L_{sc(X)}^{2}}^{2})^{1/2} , r < −1/2,

where B′ is as in (3.13).

Proof. We may choose A as above such that, in addition, i[P, A] ∈ Ψ_{sc}^{−∞,1}(X) is of the form G^{2} + E, G ∈ Ψ_{sc}^{−∞,1/2}(X), E ∈ Ψ_{sc}^{−∞,2}(X). We can take, for example, A with σ(A) = χ(ν)ϕ^{2}(p), p = σ(P), with ϕ ∈ C_{c}^{∞}(ℝ) identically 1 near λ, χ ∈ C^{∞}(ℝ) identically 0 on (−∞, −a(λ)/2), 1 on (a(λ)/2, +∞), and χ′ ≥ 0 with (χ′)^{1/2} ∈ C^{∞}(ℝ). Then the principal symbol of i[P, A] is W σ(A), and (W σ(A))^{1/2} is real and C^{∞} by (1.27), so i[P, A] = G^{2} + E as above. Now

B(u, u) = |⟨[P − λ, A]u, u⟩| ≥ ∥Gu∥^{2} − |⟨u, Eu⟩| ≥ ∥Gu∥^{2} − C∥u∥_{x^{−1}L^{2}}^{2},

so by Lemma 12.8, ∥Gu∥^{2} + ∥u∥_{x^{−1}L^{2}}^{2} ≤ C′B(u, u).

Conversely,

B(u, u) = |⟨[P − λ, A]u, u⟩| ≤ C′′(∥Gu∥_{x^{−1/2}L^{2}} + ∥u∥_{x^{−1}L^{2}})^{2},

and G may be chosen so as to satisfy the conditions of (3.13). □

Corollary 12.10. For λ ∉ Cv(V), q ∈ Min_{+}(λ), the Poisson operators

P_{+}(q, λ) : S(ℝ) → E_{ess}^{∞}(λ)
P_{+}(q, λ)(g) = u ⟺ u ∈ E_{ess}^{∞}(λ), M_{+}(q, λ)u = g, M_{+}(q′, λ)u = 0 for q′ ≠ q

extend to continuous linear maps

P_{+}(q, λ) : L^{2}(ℝ) → E_{ess}^{0}(λ).

Proof. By Proposition 12.9, for g ∈ S(ℝ),

\begin{matrix}∥P (q, λ)g∥_{3}^{2} ≤ C|B(P (q, λ)g, P (q, λ)g)| \\ = 2\sqrt{λ − V (π(q))} ∫_{ℝ} |M_{+}(q, λ)(P (q, λ)g)|^{2} ω_{q} = 2\sqrt{λ − V (π(q))} ∫_{ℝ} |g|^{2} ω_{q}, \end{matrix}

so the density of S(ℝ) in L^{2}(ℝ) finishes the proof. □

The combination of these results now yields the following theorem.

Theorem 12.11 (Theorem I.3). For λ ∉ Cv(V), the joint map (12.20)

M_{+}(λ) = E_{ess}^{0}(λ) → ⊕_{q∈Min_{+}(λ)}L^{2}(ℝ), M_{+}(λ)u = (M_{+}(q, λ)(u))_{q∈Min_{+}(λ)}

is an isomorphism with inverse

P_{+}(λ) : ⊕_{q∈Min_{+}(λ)}L^{2}(ℝ) → E_{ess}^{0}(λ), P_{+}(λ)(g_{q})_{q∈Min_{+}(λ)} = ∑_{q} P_{+}(q, λ)g_{q}

We can also express the pairing (12.15) in terms of the expansions of microlocal eigenfunctions u ∈ E_{mic,−}(q, λ), q ∈ Min_{−}(λ), i.e. in the incoming region. That leads to maps M_{−}(q, λ) and P_{−}(q, λ), for q ∈ Min_{−}(λ) and Theorem 12.11 holds with all plus signs changed to minus signs. It is also convenient to change the notation slightly and identify the spaces L^{2}(ℝ) for the incoming and outgoing radial points over z ∈ Min.

Corollary 12.12. For λ ∉ Cv(V), the S-matrix may be identified as the unitary operator S(λ) = M_{+}(λ)P_{−}(λ) on ⊕_{z∈Min}L^{2}(ℝ).

This theorem is essentially a pointwise version of asymptotic completeness in λ. Integrating in λ gives a version of the usual statement. Namely, let I ⊂ ℝ\Cv(V) be a compact interval. For f ∈ Ċ^{∞}(X) orthogonal to E_{pp}(I), u = u(λ) = R(λ + i0)f, as discussed above,

\begin{aligned}B(u, u) = B_{λ}(u(λ), u(λ)) & = i (⟨f, R(λ + i0)f⟩ − ⟨R(λ + i0)f, f⟩) \\ & = i⟨f, [R(λ + i0) − R(λ − i0)]f⟩ = 2π⟨f, Sp(λ)f⟩. \end{aligned} (12.21)

Integrating over λ in I, denoting the spectral projection of P to I by Π_{I}, and writing M_{+}(q, λ) = 0 if V (z) > λ, z = π(q), we deduce that

(12.22) ∥Π_{I}f∥^{2} = ∑_{q∈Min_{+}(λ)} 4π\sqrt{λ − V (z)} ∫ ∥M_{+}(q, λ)R(λ + i0)f∥_{L^{2}(ℝ),ω_{q}}^{2} dλ,

so M_{+} ∘ R(. + i0) is an isometry on the orthocomplement of the finite dimensional space E_{pp}(I) in the range of Π_{I}.

Theorem 12.13 (Asymptotic completeness). If I ⊂ ℝ \ Cv(V) is compact then

M_{+} ∘ R(. + i0) : Ran(Π_{I}) ⊖ E_{pp}(I) → L^{2}(I × ℝ)

is unitary.

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Centre for Mathematics and its Applications, Australian National University, Canberra ACT 0200 Australia

   E-mail address: hassell@maths.anu.edu.au

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

   E-mail address: rbm@math.mit.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

   E-mail address: andras@math.mit.edu