GEOMETRIC OPTICS AND THE

BOTTOM OF THE SPECTRUM

Richard B. Melrose

Massachusetts Institute of Technology

Abstract. The continuous spectrum, near its infimum, of the Laplacian for certain classes of ‘doubly-warped’ complete Riemannian metrics on the interiors of compact manifolds with boundary is examined using geometric optics. General conditions on the geometry of an algebra of vector fields leading to a calculus of pseudodifferential operators are described and, once verified in this particular case, the associated notion of wavefront set can be used to show the existence of an absolute scattering matrix. This operator is conjectured to be a Fourier integral operator; the confirmation of this conjecture is discussed in the special case of a ‘double-zero’ metric where the fibrations are trivial.

Introduction

In this note the structure of the bottom part, which is to say near 0, of the spectrum of the Laplacian (on functions) is discussed, for a complete metric on the interior of a compact manifold with boundary exhibiting a ‘doubly-warped’ structure near the boundary. Namely if X is the compact manifold with boundary and ∂X is its boundary, then it is assumed that there is a two-level fibration of the boundary

∂X ⟶^{ϕ_{1}} Y_{1} ⟶^{ϕ_{2}} Y_{2}, ψ = ϕ_{2} ∘ ϕ_{1}.

Here both ϕ_{1} : ∂X ⟶ Y_{1} and ϕ_{2} : Y_{1} ⟶ Y_{2} are fibrations of compact manifolds. By taking a product decomposition of X near its boundary these fibrations can be extended to a neighbourhood of the boundary; denote such a product extension by

[0, δ) × ∂X ⟶^{ϕ̃_{1}} [0, δ) × Y_{1} ⟶^{ϕ̃_{2}} [0, δ) × Y_{2}, ψ̃ = ϕ̃_{2} ∘ ϕ̃_{1}.

Consider a metric on the interior of X which, for some such choice of extensions takes the form

(1)  g = \frac{ψ̃^{∗}h̃_{2}}{x^{4}} + \frac{ϕ̃_{1}^{∗}h̃_{1}}{x^{2}} + h̃_{0}, near ∂X.

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Here, h̃_{2}, h̃_{1} and h̃_{0} are, respectively, metrics on [0, δ)×Y_{2}, [0, δ)×Y_{1} and X and it is also assumed that, near ∂X, x = ψ̃^{∗}x̃ where x̃ is the distance from the boundary in the h̃_{2} metric. In fact it is enough that h̃_{1} and h̃_{0} be smooth symmetric 2-cotensors which restrict to metrics on the fibres of ϕ_{2} and on the fibres ψ respectively. Under these assumptions near the boundary g, further assumed to be a Riemannian metric on the interior of X, is necessarily complete.

The Laplacian associated to g, Δ, is an essentially self adjoint operator on the domain Ċ^{∞}(X) ⊂ C^{∞}(X) consisting of the smooth functions vanishing to all orders at the boundary. The general structure of the spectrum of such an operator can be described as follows.

Theorem 1. For any metric of the type (1), with dim Y_{1} + dim Y_{2} > 0, the essential spectrum of the Laplacian on functions is absolutely continuous of infinite multiplicity on [0, ∞).

In this theorem I do not exclude the possibility of embedded discrete spectrum. In all likelihood it does not occur at all if the fibres of ϕ_{1} have dimension zero and in general there should be no embedded spectrum near 0 assuming dim Y_{1} + dim Y_{2} > 0. If it occurs, the eigenfunctions corresponding to such discrete spectrum are in the space Ċ^{∞}(X).

Although this gives a reasonably full description of the spectrum of Δ, a more detailed description of the generalized eigenfunctions is possible through the approach of ‘geometric’ scattering theory. This can be described in a variety of ways. One rather transparent approach is through a generalized (or degenerate) boundary problem.

Theorem 2. For a metric of type (1), if λ ∈ ℝ, 0 < λ^{2} < A^{2}, where A^{2} is the infimum of the positive eigenvalues of the Laplacian on the fibres of ϕ_{1}, then for each f ∈ C^{∞}(Y_{1}) there is a unique generalized eigenfunction with eigenvalue λ^{2} of the form (2)

\begin{matrix}u = P_{λ}(f) = e^{iλ/x}x^{q}u_{+} + e^{−iλ/x}x^{q}u_{−}, u_{±} ∈ C^{∞}(X), u_{+} ↿ ∂X = ϕ_{1}^{∗}f, \\ q = \frac{1}{2} dim Y_{1} + dim Y_{2}, \end{matrix}

orthogonal in L^{2} to any eigenfunctions in Ċ^{∞}(X) with eigenvalue λ^{2}.

The proof of this result is outlined below, with particular emphasis on the newer aspects.

The map whose existence is asserted by this Proposition is the Poisson operator for the generalized boundary value problem at ‘wave number’ λ. Since the Laplacian is real, the uniqueness part of this result shows that

P_{−λ}f = \overline{P_{λ}\overline{f}}.

For this reason it is enough to suppose throughout that λ > 0. Furthermore the absolute scattering operator can be defined by

A_{λ} : C^{∞}(Y_{1}) ⟶ C^{∞}(Y_{1}), A_{λ}f = f_{−}, u_{−} ↿ ∂X = ϕ_{1}^{∗}f_{−},

where u_{±} are as in (2); this operator is necessarily invertible. Its microlocal properties are described here in some cases and the structure in the general case is conjectured.

Let me give (rather ad hoc) nomenclature for some of the special cases of the metrics in (1) and hence discuss some results previously established.

First, in case ϕ_{1} is the identity and Y_{2} is a point, so h̃_{2} = dx̃^{2} and h̃_{0} is absent, we arrive at the notion of a scattering metric considered in [8] and [10].

Theorem 3. [10] In the case of a scattering metric (ϕ_{1} has trivial fibres, ϕ_{2} has one fibre) A_{λ} is a Fourier integral operator associated with geodesic flow at time π for the induced metric on the boundary.

The case in which dim Y_{2} = 0 but ϕ_{1} can have non-trivial fibres is described in [8] and in more detail in [3] where such metrics are called ‘fibred cusp’ metrics. The same result is shown there in this case:

Theorem 4. [3] For a fibred cusp metric the scattering operator, for 0 < λ^{2} < A^{2}, is a Fourier integral operator on C^{∞}(Y_{1}) associated with geodesic flow at time π, for the metric induced by h̃_{1} on Y_{1}.

In the special case where ϕ_{1} and ϕ_{2} are isomorphisms, so it can be assumed that Y_{2} = ∂X, the metric can be described as a ‘double-zero’ metric (because the associated Lie algebra of vector fields, discussed below, consists of the vector fields which vanish to second order at the boundary). This is the main new case discussed here. The integration of this case with the ‘fibred cusp’ case can be expected to allow the full analysis of the general doubly-warped metric in (1).

In the case of a double-zero metric the construction of a parametrix for the Poisson operator is outlined here. Carried to its proper conclusion this yields

Theorem 5. If h is a non-singular Riemann metric on a compact manifold with boundary X and x ∈ C^{∞}(X) is equal to the distance from the boundary up to a quadratic term (i.e. modulo x^{2}C^{∞}(X) near the boundary) then the absolute scattering operator defined in (2) is a pseudodifferential operator.

Formally combining this result and that from [10] one can give a conjecture in the general case. The identity wavefront relation for the scattering operator implied by Theorem 5 can be understood in terms of the limiting behaviour of complete geodesics on X which tend uniformly to the boundary. This limiting geodesic relation can be analyzed in terms of the rescaled symplectic geometry which arises from the (formal part of) the algebra of pseudodifferential operators discussed below. In this way the various boundary variables in (1) can be seen to move at ‘different speeds’ according to the associated powers of x. The variables in the fibres of ϕ_{1} are ‘fast’ and the problem is global in these variables. The variables in the fibres of ϕ_{2} are ‘finite speed’ whilst those in Y_{2} are ‘slow’. The limiting geodesic relation for a doubly-warped metric descends to a relation on T^{∗}Y_{1} in which motion is along the fibres, with the distance scaled from π to 0 according to the length of the projection of the initial covector into the fibre normal. Using this formal reasoning (described also in [7]) one can expect the following.

Conjecture. If g is a doubly-warped Riemann metric on a compact manifold with boundary X and 0 < λ^{2} < A^{2} the absolute scattering operator A_{λ}, defined in (2), is a Fourier integral operator associated to the limiting geodesic relation at infinity.

The initial step in the proof of the existence of generalized eigenfunctions as in (2) is the construction of a suitably tailored algebra of pseudodifferential operators, with its associated wavefront set. I shall use this opportunity to outline what I believe is the ‘appropriate’ structure to give such an algebra and also a related ‘large’ calculus, this is done in §1. The doubly-warped metric is then shown, in §2, to correspond to a Lie algebra of vector fields satisfying these general conditions. In the particular case of double-zero metrics the calculus so defined, and its associated wavefront set, are used in §3 to show the existence of generalized eigenfunctions; the general case should follow by combining the methods here with those of [3]. In §4 a more constructive approach is taken, as in [10], which leads to a a parametrix for the Poisson operator and hence a proof of Theorem 5. The same approach in the general setting should yield the Conjecture.

The author thanks the organizers of the meeting at Cortona for the opportunity to speak there and would also like to acknowledge discussions with Rafe Mazzeo, who also kindly commented on the manuscript, and Andras Vasy, who suggested an alternative proof of a slightly weaker form of Theorem 5 and the Conjecture which restored the author’s faith!

§1: Boundary structures

To construct an algebra of pseudodifferential operators appropriate to the analytic questions of interest here a general procedure of ‘microlocalization’ will be invoked. This is based on the differential analysis of manifolds with corners and I shall therefore describe some general definitions and results. These should eventually appear in [9]; in the meantime they can be extracted from various places [1], [4], [5], [6], [7].

Manifolds with corners. An intrinsic definition of a manifold with corners is as a (connected) Hausdorff, paracompact topological manifold with boundary X with a ring of functions C^{∞}(X) which can be embedded as a submanifold with corners of a smooth manifold of the same dimension. That is, there is a continuous map i : X ⟶ X̃ into a C^{∞} manifold X̃, such that i is a homeomorphism onto its image, C^{∞}(X) = i^{∗}C^{∞}(X̃) and there are k non-negative functions ρ_{i} ∈ C^{∞}(X̃), i = 1, …, k such that

i(X) = {p ∈ X̃; ρ_{i}(p) ≥ 0, i = 1, …, k} and
∀ p ∈ X̃, {dρ_{i}(p); ρ_{i}(p) = 0} is an independent set.

This is the same as a coordinate-covering definition, with C^{∞} transitions, based on the local model space [0, ∞)^{r} × ℝ^{n−r} (for fixed n but variable r) except that it ensures that the boundary hypersurfaces, which are the components of the sets i^{−1}({ρ_{i} = 0}), are embedded. These components are each manifolds with corners with the C^{∞} structure given by restriction of C^{∞}(X). The set of these boundary hypersurfaces is denoted M_{1}(X); each of these necessarily has a defining function ρ_{H} ∈ C^{∞}(X), H ∈ M_{1}(X), such that

ρ_{H} ≥ 0, H = {ρ_{H} = 0} and dρ_{H}(p) ≠ 0 ∀ p ∈ H.

Here df(p) ∈ T_{p}^{∗}X is an element of the intrinsic cotangent bundle which is identified with T_{i(X)}^{∗}X̃ by i, i.e. it is the pull-back under i of T^{∗}X̃. Similar definitions apply to any of the usual natural bundles over a manifold without boundary.

p-submanifolds. There are various notions of submanifold of a manifold with corners, the most restrictive of which is a p-submanifold where the ‘p’ stands for product. Notice that a manifold with corners always has a product decomposition near each point p, in the sense that p has a neighbourhood O_{p} which is diffeomorphic to a product [0, 1)^{k} × (−1, 1)^{n−k}. A closed subset Y ⊂ X is a p-submanifold if each point p ∈ Y has such a neighbourhood O_{p} with a diffeomorphism which also (i.e. simultaneously) reduces Y ∩ O_{p} to a product {0}^{r} × [0, 1)^{k−r} × {0}^{s} × (−1, 1)^{n−k−s}. It follows that Y itself is a manifold with corners. b-maps. A C^{∞} map f : X ⟶ Y, between manifolds with corners is defined by the usual condition that f^{∗}C^{∞}(Y) ⊂ C^{∞}(X). It is useful to deal with a restricted class of smooth maps which ‘respect the boundaries’. Namely consider the notion of an interior b-map defined by the requirement in terms of boundary defining functions that

\begin{matrix}f^{∗}ρ′_{G} = a_{G} ∏_{H∈M_{1}(X)} ρ_{H}^{e(H,G)} = a_{G}ρ^{e(G)}, \\ 0 < a_{G} ∈ C^{∞}(X), ∀ G ∈ M_{1}(Y). \end{matrix}

Since f is assumed smooth, the boundary exponents e(H, G) are nonnegative integers for each H ∈ M_{1}(X) and G ∈ M_{1}(X). Note that e(H, G) is positive if and only if f(H) ⊂ G. The obvious multiindex notation is used here to write the powers of boundary defining functions with e(G) : M_{1}(X) ⟶ ℕ_{0} the ‘multiindex’.

A general b-map is an interior b-map into one of the boundary faces of the range space.

Tangent vector fields. The smooth vector fields on X, meaning the smooth sections of T X, form a Lie algebra. A more significant Lie subalgebra is the space, V_{b}(X), of smooth vector fields tangent to each boundary hypersurface. This has many properties similar to the whole Lie algebra. In particular it is the space of sections of a smooth vector bundle over X,

V_{b}(X) = C^{∞}(X;^{b}T X).

This new bundle is bundle-isomorphic to T X but not in any natural way. There is a natural C^{∞} vector bundle map

ι_{b} : ^{b}T X ⟶ T X

which is a diffeomorphism over the interior and has fibre rank exactly k at any boundary point of codimension k, i.e. point which lies in precisely k boundary hypersurfaces. The null space of ι_{p} at such a point is a vector space, ^{b}N_{p}X, which is naturally the product of k oriented lines, each corresponding to one of the hypersurfaces passing through p.

b-differential. By continuity from the usual differential in the interior, an interior b-map, as defined above, has both a push-forward and a pull-back b-differential (dual to each other)

bf_{∗} : ^{b}T_{p}X ⟶ ^{b}T_{f(p)}Y and ^{b}f^{∗} : ^{b}T_{f(p)}^{∗}Y ⟶ ^{b}T_{p}^{∗}X.

It follows automatically that at any boundary point

bf_{∗} : ^{b}N_{p}X ⟶ ^{b}N_{f(p)}Y.

Using these maps, two surjectivity properties for an interior b-map can be identified. Namely f is said to be a b-submersion if it is surjective and has surjective b-differential at each point. On the other hand it is said to be b-normal if at each boundary point the restricted bdifferential on the b-normal spaces is surjective. If it is both b-normal and is a b-submersion it is said to be a b-fibration.

Each of the four classes just defined, interior b-maps, b-submersions, b-normal maps and b-fibrations is closed under composition. Some of their analytic properties are described below.

Generalized blow-down maps. Another class of b-maps that typically arise from blow-up constructions have a weaker surjectivity property that b-submersions. Namely a b-map between manifolds with corners of the same dimension is said to be a generalized blow-down map if it restricts to a diffeomorphism of the interiors and its pull-back differential has the property

C^{∞}(X) ·^{b}f^{∗}C^{∞}(Y;^{b}T^{∗}Y) ⊃ ρ^{a}C^{∞}(X;^{b}T^{∗}X) for some multiindex a.

This implies in particular that if ν_{b}(Y) is a positive b-density on Y then pull-back of densities induces a map of the form

f^{∗}ν_{b}(Y) = aρ^{w}ν_{b}(X), w : M_{1}(X) ⟶ ℕ_{0}, 0 < a ∈ C^{∞}(X).

It further follows that a generalized blow-down map always induces an isomorphism on functions and densities

f^{∗} : Ċ^{∞}(Y) ⟷ Ċ^{∞}(X), f_{∗} : Ċ^{∞}(X; Ω) ⟷ Ċ^{∞}(Y, Ω).

This in turn implies that it induces an isomorphism from the space of extendible distributional sections of any bundle

f^{∗} : C^{−∞}(Y; G) ⟷ C^{−∞}(X; f^{∗}G), C^{−∞}(Y; G) = (Ċ^{∞}(Y; G′ ⊗ Ω))′ .

Thus a generalized blow-down map can be thought of as a ‘distributional isomorphism’ in the sense that it changes the structure near the boundary but only to finite order. Notice that the composite of two generalized blow-down maps is again a generalized blow-down map.

The generalized blow-down maps considered here are constructed from radial blow-down maps. Part of the construction shows that the blow down map for blow up of a boundary p-submanifold, i.e. a psubmanifold of a proper boundary face, is always a generalized blowdown map in this sense.

Lifting. If f : X ⟶ Y is a generalized blow-down map let K ⊂ Y be the ‘singular set’ which is the image of the set near which it is not locally a diffeomorphism. Then the lift of a subset C ⊂ Y is defined provided either C is the closure of its intersection with the complement of K i.e. C = Clos(C ∩ (Y \ K)), when by definition it is the set

f^{∗}(C) = Clos(f^{−1}(C ∩ (Y \ K))

or in case C ⊂ K when it is taken to be

f^{∗}(C) = f^{−1}(C).

This definition reflects the commutation properties of blow ups.

Boundary structure. By a boundary structure (this is a special case of the more general notion of a boundary-fibration structure) on a compact manifold with corners X, I shall mean a Lie subalgebra W ⊂ V_{b}(X) which can be ‘resolved’ in a sense made precise in terms of the notions introduced above. These conditions are just formal properties of the products X^{2} and X^{3} in their relation to X in the boundaryless case.

(1) There must be two manifolds with corners X_{W}^{2} and X_{W}^{3} and generalized blow-down maps

β^{2} : X_{W}^{2} ⟶ X^{2}, β^{3} : X_{W}^{3} ⟶ X^{3}

such that the symmetry groups X^{2} ∋ (x, y) ⟶ (y, x) ∈ X^{2} and the similar exchange maps on X^{3} lift to diffeomorphisms of X_{W}^{2} and X_{W}^{3} respectively.

(2) Under β^{2} the diagonal Δ = {(x, x) ∈ X^{2}; x ∈ X}, lifts to a p-submanifold Δ_{W} ⊂ X_{W}^{2} such that β^{2} restricts to a diffeomorphism of Δ_{W} to Δ.

(3) The two composite maps

π_{O,W} = π_{O} ∘ β_{W}^{2}, O = R, L

are b-fibrations, where π_{R} and π_{L} are the right and left projections from X^{2} to X

(4) The elements of W, lifted to X^{2} as vector fields on either the left or right factor, lift under β^{2} into V_{b}(X_{W}^{2}) to give a space of vector fields transversal to Δ_{W}.

(5) Let π_{O} for O = S, C, F be the three projections π_{O} : X^{3} ⟶ X^{2} corresponding, respectively, to dropping the left, the central and the right factor of X. Then there are required to be three b-fibrations each giving a commutative diagramme

\begin{matrix}X_{W}^{3} & ⟶^{π_{O,W}} & X_{W}^{2} \\ ^{β^{3}}↓ & & ↓^{β^{2}} \\ X^{3} & ⟶_{π_{O}} & X^{2}, \end{matrix} O = F, C, L,

such that under π_{O,W} the submanifold Δ_{W} ⊂ X_{W}^{2} lifts to a p-submanifold of Δ_{O,W} ⊂ X_{W}^{3} to which the other two π_{O′,W}, {S, C, F } ∋ O′ ≠ O restrict to diffeomorphisms.

Given the symmetry condition it suffices to check the last two conditions for one case. Over the interiors of X, X^{2} and X^{3} all these conditions are trivial except that they ensure that W must be unconstrained there. At the boundary the existence of such a resolution is distinctly non-trivial. Even in the simplest of cases, such as for V_{b}(X) described next, such a resolution need not be unique.

One property which follows from these conditions is that W = C^{∞}(X;^{W}T X) for a vector bundle ^{W}T X over X.

Notation. Essentially by definition a boundary structure arises from some sort of ‘uniformity’ structure near the boundary. The Lie algebra of vector fields is generally denoted V_{n}(X) for some abbreviated name n. Then the spaces and maps are denoted X_{n}^{2}, β_{n}^{2} etc.; note that the ‘name’ refers to the resolution rather than just the space of vector fields.

Basic example, V_{b}(X). For any compact manifold with corners the space of all tangent vector fields V_{b}(X) is a boundary structure on X. This is not however completely trivial! In particular it is certainly not the case that the double space X_{b}^{2} is the product X^{2}, rather it is obtained from this product by blowing up all the codimension two boundary faces H × H where H ∈ M_{1}(X). The triple product X_{b}^{3} is similarly defined from X^{3} by first blowing up all the triple products, H^{3} ⊂ X^{3} and then all the double products of the form X × H^{2}, H × X × H or H^{2} × X.

Even in the case of a manifold with boundary this resolution is not always unique. If the boundary has two components, H_{1}, and H_{2} ∈ M_{1}(X) then one can also define an ‘overblown’ resolution with X_{ob}^{2} defined by blowing up H_{1} × H_{2} and H_{2} × H_{1} in addition to H_{1}^{2} and H_{2}^{2}.

Conormal distributions at a p-submanifold. Conormal distributions with respect to an interior p-submanifold (one that meets the interior) can be defined just as in the original case of an embedded submanifold of a manifold without boundary as considered by Hörmander in [2]. Indeed it is always possible to ‘double’ a manifold with corners successively across each boundary hypersurface in such a way that a given p-submanifold extends to a closed embedded submanifold of an extension X̃. Then conormal distributions with respect to an interior p-submanifold Y ⊂ X are just the restrictions of conormal distributions for the extension Ỹ ⊂ X̃. The resulting spaces are independent of the choice of extension and their properties can be deduced from the boundaryless case. The space of conormal distributions with respect to a p-submanifold can then be denoted I^{m}(X; Y). The 1-step polyhomogeneous subspace I_{phg}^{m}(X; Y) is similarly well defined, as are the corresponding spaces of sections of any vector bundle. These spaces are always C^{∞}(X) modules with principal symbol maps as in the boundaryless case.

Conormal coefficients. As well as the natural space, C^{∞}(X), of smooth functions on a manifold with corners it is frequently useful to consider larger spaces of distributions conormal with respect to the boundary. Since the boundary may consist of a number of intersecting hypersurfaces these distributions are somewhat more complicated than in the case of a p-submanifold. However, the ‘spaces with bounds’ are easily defined. Let ν_{b} be a smooth positive b-density on X, assumed compact for convenience. Thus if

ρ_{tot} = ∏_{H∈M_{1}(X)} ρ_{H}

is a ‘total boundary defining function’ then 0 < ρ_{tot}ν_{b} ∈ C^{∞}(X; Ω) is a smooth positive measure. Now, define

L_{b}^{2}(X) = {f ∈ L_{loc}^{2}(X^{∘}); ∫_{X} |f|^{2}ν_{b} < ∞}

and more generally if α : M_{1}(X) ⟶ ℝ is a ‘multiindex’ set

ρ^{α}L_{b}^{2}(X) = {f ∈ L_{loc}^{2}(X^{∘}); ∫_{X} |ρ^{−α}f|^{2}ν_{b} < ∞}

where ρ^{α} = ∏_{H∈M_{1}(X)} ρ_{H}^{α(H)}.

Since L_{loc}^{2}(X^{∘}) is a well-defined subspace of the space of distributions on the interior, derivatives can be interpreted distributionally and then

ρ^{α}H_{b}^{∞}(X)

= {f ∈ ρ^{α}L_{b}^{2}(X); V_{1} … V_{l}f ∈ ρ^{α}L_{b}^{2}(X), ∀ V_{i} ∈ V_{b}(X), ∀ l} .

GEOMETRIC OPTICS AND THE BOTTOM OF THE SPECTRUM 11

These are the L^{2}-based conormal distributions (or functions since they are necessarily C^{∞} in X^{∘}.) There are similar L^{∞}-based conormal functions, obtained by replacing L_{b}^{2}(X) by L^{∞}(X) throughout. These are more naturally thought of as symbols so we set

S^{α}(X)

= {f ∈ ρ^{−α}L^{∞}(X); V_{1} … V_{l}f ∈ ρ^{−α}L^{∞}(X), ∀ V_{i} ∈ V_{b}(X), ∀ l} .

A form of the Sobolev embedding theorem then gives the ‘Sandwich Theorem’

S^{−α−ϵ}(X) ⊂ ρ^{α}H_{b}^{∞}(X) ⊂ S^{−α}(X) ∀ α

provided ϵ is a positive multiindex, i.e. all entries are strictly positive.

Polyhomogeneity. As well as these conormal distributions defined by bounds there are polyhomogeneous space. By definition an index set is a subset

E ⊂ ℂ × ℕ_{0}, ℕ_{0} = {0, 1, 2, … }

satisfying the following four conditions

E is discrete (z_{l}, k_{l}) ∈ E, l ∈ ℕ, |z_{l}| + k_{l} → ∞ ⟹ Re(z_{l}) → ∞

(z, k) ∈ E, l ∈ ℕ_{0}, 0 ≤ l ≤ k ⟹ (z, l) ∈ E
(z, k) ∈ E, r ∈ ℕ ⟹ (z + r, k) ∈ E.

An index family 𝔈 for X is an assignment of an index set 𝔈(H) to each H ∈ M_{1}(X).

If 𝔈 is an index family for X then the space of polyhomogeneous conormal distributions, A^{𝔈}(X), at the boundary with powers determined by 𝔈 can be defined so that it contains the functions

∏_{H∈M_{1}(X)} ρ_{H}^{z(H)} log^{k(H)} ρ_{H},

provided (z(H), k(H)) ∈ 𝔈(H) and in addition has appropriate completeness properties. These spaces are modules over C^{∞}(X) so one can also define

A^{𝔈}I^{m}(X; Y) = A^{𝔈}(X) ˆ⊗_{C^{∞}(X)}I^{m}(X; Y)

as appropriate completions of the tensor products. A similar argument allows the corresponding spaces of sections of any vector bundle over X to be defined.

Kernels and operators. The space of pseudodifferential operators on a compact manifold without boundary, acting on half-densities, is identified with the corresponding space of kernels

Ψ^{m}(X; Ω^{\frac{1}{2}} ) = I^{m}(X^{2}, Δ; Ω^{\frac{1}{2}} ).

In the case of a compact manifold with corners an appropriate form of the Schwartz kernel theorem identifies all continuous linear operators

A : Ċ^{∞}(X; Ω^{\frac{1}{2}} ) ⟶ C^{−∞}(X; Ω^{\frac{1}{2}} )

with their kernels which are arbitrary elements of C^{−∞}(X^{2}; Ω^{\frac{1}{2}} ). The assumption that β^{2} : X_{W}^{2} ⟶ X^{2} is a generalized blow-down map implies that the identification over the interior extends by continuity to give an isomorphism

(β^{2})^{∗} : C^{−∞}(X^{2}; Ω^{\frac{1}{2}} ) ⟷ C^{−∞}(X_{W}^{2}; Ω^{\frac{1}{2}} ).

Thus a space of operators on X can be defined by specifying the kernels as half-densities on X_{W}^{2}.

The identity. The relation between X_{W}^{2} and X demanded of a resolution implies that β^{2} induces a diffeomorphism from the lifted diagonal to the diagonal of X^{2}. From this, and the assumption that it is a generalized blow-down map, it follows that the kernel of the identity operator, which is a non-vanishing Dirac section of Ω^{\frac{1}{2}} for the diagonal, lifts to a distribution

Id ∈ C^{−∞}(X_{W}^{2}; Ω^{\frac{1}{2}} ), Id = ρ^{κ}A, A ∈ I^{0}(X_{W}^{2}, Δ_{W}; Ω^{\frac{1}{2}} ), A > 0.

That is, apart from a boundary weight ρ^{κ} it too is a positive Dirac section of the half-density bundle for the lifted diagonal. The index κ is only determined for the boundary hypersurfaces of X_{W}^{2} which meet the lifted diagonal (they are in one to one correspondence with the boundary hypersurfaces of X).

W-pseudodifferential operators. Using the assumption that the Lie algebra W is transversal to the lifted diagonal it follows that the algebra of differential operators on half-densities generated by W and C^{∞}(X) is identified with the space

\{A ∈ I^{ℕ_{0}}(X_{W}^{2}, Δ_{W}; ρ^{κ}Ω^{\frac{1}{2}} ); supp(A) ⊂ Δ_{W}\} .

That is, the space of W-differential operators is identified with the space of all smooth Dirac sections over the lifted diagonal of the blown-up space X_{W}^{2}.

This identification of the W-differential operators makes it natural to define the ‘small’ calculus of W-pseudodifferential operators in terms of kernels by

\begin{matrix}Ψ_{W}^{m}(X; Ω^{\frac{1}{2}} ) = \\ \{A ∈ I^{m}(X_{W}^{2}, Δ_{W}; ρ^{κ}Ω^{\frac{1}{2}} ); A ≡ 0 at H ∈ M_{1}(X_{W}^{2}) if H ∩ Δ_{W} = ∅\} . \end{matrix}

Larger spaces of W-pseudodifferential operators can be defined for any choice of index set 𝔈 for X_{W}^{2} by

Ψ_{W}^{m,𝔈}(X; Ω^{\frac{1}{2}} ) = A^{𝔈}I^{m}(X_{W}^{2}, Δ_{W}; ρ^{κ}Ω^{\frac{1}{2}} ).

If 𝔈 satisfies the compatibility condition that 𝔈(H) ⊂ ℕ_{0} × {0} if H ∩ Δ_{W} ≠ ∅, then C^{∞}(X_{W}^{2}) ⊂ A^{𝔈}(X_{W}^{2}) and then

Diff_{W}^{m}(X; Ω^{\frac{1}{2}} ) ⊂ Ψ_{W}^{m}(X; Ω^{\frac{1}{2}} ).

These definitions can be extended to operators acting on sections of any vector bundle(s).

Pull-back. Conormal functions can be pulled back under general interior b-maps provided they have a transversality property with respect to the p-submanifold involved (if there is one). Since it suffices for present purposes, attention can be restricted here to b-fibrations. Then each boundary hypersurface H ∈ M_{1}(X) is mapped onto either the whole of Y or onto a boundary hypersurface G = F (H) ∈ M_{1}(Y). If 𝔈 is an index family for Y define 𝔉 = f^{#}𝔈 by setting 𝔉(H) = ℕ × {0} (the index set of C^{∞}) in the first case and

𝔉(H) = {(z′, k); z′ = e(H, G)z, (z, k) ∈ 𝔈(G)}

in the second case.

Proposition. Suppose f : X ⟶ Y is a b-fibration of compact manifolds with corners and that B ⊂ Y is an interior p-submanifold then f^{∗}(B) ⊂ X is a p-submanifold and pull back defines linear maps

f^{∗} : A^{𝔈}I_{phg}^{m}(Y, B; G) ⟶ A^{𝔉}I_{phg}^{m′}(X, f^{∗}B; f^{∗}G), 𝔉 = f^{#}𝔈

here f^{∗}G is the pull-back of the bundle G and m′ − m depends on the relative dimensions.

This result follows from a direct characterization of the polyhomogeneous conormal functions in terms of expansions.

Push-forward. There is a similar result for push-forward. For an index set E write E > 0 to mean that (z, k) ∈ E ⟹ Re z > 0. For a b-fibration f : X ⟶ Y and an index set 𝔈 on X define an index set 𝔉 = f_{#}𝔈 as follows. For each G ∈ M_{1}(Y) set

𝔉(G) = {(z, k); ∃ (z_{i}, k_{i}) ∈ 𝔈(H_{i}), F (H_{i}) ⊂ G,

z = ∑_{i} \frac{z_{i}}{e(H_{i}, G)}, k + 1 ≤ ∑_{i}(k_{i} + 1)\}.

Proposition. Suppose f : X ⟶ Y is a b-fibration of compact manifolds with corners and B ⊂ X is an interior p-submanifold to which f is (b-)normal, then f(B) ⊂ Y is an interior p-submanifold and provided 𝔈(H) > 0 for all H ∈ M_{1}(X) mapped onto Y push-forward defines linear maps

f_{∗} : A^{𝔈}I_{phg}^{m}(X, B; f^{∗}G ⊗ Ω_{b}) ⟶ A^{𝔉}I_{phg}^{m′}(X, f(B); G ⊗ Ω_{b}), 𝔉 = f_{#}𝔈;

here m′ − m depends on the relative dimensions.

This has further extensions to the case, which occurs in the composition of pseudodifferential operators, where B is replaced by two transversally intersecting submanifold, each of which is mapped onto Y but the intersection of the two is mapped onto a proper p-submanifold. A similar result then holds for spaces of conormal distributions defined with respect to the intersecting pair. The important point for the present discussion is that these contain the products of conormal distributions associated to the two submanifolds; again this is just as in the boundaryless case.

Operators. As discussed above the kernels of these W-pseudodiffer-ential operators are defined on X_{W}^{2} and hence equivalently as extendible distributions on X^{2}. They therefore define operators from Ċ^{∞}(X) to C^{−∞}(X). For the small calculus

A ∈ Ψ_{W}^{m}(X) ⟹ A : ρ^{a}C^{∞}(X) ⟶ ρ^{a}C^{∞}(X)

for any multiindex a. Since the calculus is closed under passage to adjoints these operators act on extendible distributions. Elements of the large calculus determined by a choice of index family for X_{W}^{2} may not act on any fixed space, depending on the index families, but do map appropriately ‘integrable’ polyhomogeneous spaces to polyhomogeneous spaces with transformed index families.

Symbol and normal morphisms. The symbol of an element of the small calculus is the symbol of its kernel, as a conormal distribution. This is an equivalence class of functions (or in general sections of a bundle) on the conormal bundle to the lifted diagonal. The hypotheses above identify this bundle with the structure bundle ^{W}T^{∗}X.

Elements of the small calculus act on ρ^{a}C^{∞}(X) and hence by restriction define operators on C^{∞}(H) for each H ∈ M_{1}(X). These boundary actions are determined by the restriction of the kernels to the corresponding boundary hypersurface of X_{W}^{2}. There restricted kernels in fact contain more information and can be viewed as the kernels of W_{H}- pseudodifferential operators (with additional invariance properties) for a boundary(-fibration) structure on ^{b}NH. Thus even to handle operators associated to boundary structures one needs the more general theory including the fibration part of boundary-fibration structures. For the cases discussed below these normal operators can be handled directly.

Composition. Using the push-forward and pull-back theorems just described, and extensions as in the boundaryless case it follows that the small calculus is always an algebra containing the W-differential operators and the large calculus is a two-sided module over the small calculus. Elements of the larger calculus can only be composed under appropriate conditions on the index families but once these are satisfied the composite is in the calculus.

Sobolev spaces. There is a natural L^{2} space, L_{W}^{2}(X) defined by a density given locally by the wedge product of a basis of ^{W}T^{∗}X. Using this basic space and the action of the small calculus on distributions, Sobolev spaces can be defined in terms of weighting by powers of boundary defining functions

H_{W}^{a,m}(X) = {u ∈ C^{−∞}(X); Au ∈ ρ^{a}L2_{W}(X) ∀ A ∈ Ψ_{W}^{−m}(X)}.

Blow up. Essentially by definition the spaces X_{W}^{2} and X_{W}^{3} are defined by ‘blow-up’. The radial blow up of a p-submanifold Y of a manifold with corners X is a well-defined procedure giving a new manifold with corners [X; Y] with natural blow-down map β : [X; Y] ⟶ X. This new manifold has an additional boundary hypersurface compared to X which replaces Y. Polar coordinates in the normal variables to Y in X lift to local coordinates near this ‘front face’. If Y is a boundary p-submanifold then β is a generalized blow-down map in the sense described above.

Amongst the important properties of blow up are two commutation properties. If Y_{1} ⊂ Y_{2} are two p-submanifolds then under blow-up of either of them the other lifts to be a p-submanifold and denoting the lifts by the same letter the two blow ups are naturally isomorphic

[X; Y_{1}; Y_{2}] ≡ [X; Y_{1}; Y_{2}], Y_{1} ⊂ Y_{2} p-submanifolds.

Here the naturality is that the isomorphism gives a commutative diagram with the blow-down maps. The same result is true if Y_{1} and Y_{2} meet transversally, but not in general.

§2: The doubly-warped calculus

The ‘elliptic’ part of the analysis of the Laplacian associated to a metric as in (1) consists in the construction of an algebra of pseudodifferential operators on X which contains the resolvent away from the spectrum. In this case it is the ‘small calculus’ discussed above for the appropriate Lie algebra of smooth vector fields which describes the degeneration of the metric in the precise sense that the Laplacian is a (symbolically) elliptic operator in the corresponding enveloping algebra.

Theorem 6. The space V_{dw}(X) consisting of those smooth vector fields on X which have bounded length with respect to the metric g in (1), is a Lie algebra which is a boundary structure in the sense of the previous section.

Proof. The space V_{dw}(X) is easily described. Local coordinates near any boundary point of the form x, z, y, t can be introduced with x being the distance function x̃ lifted from [0, δ) × Y_{2}, the t’s are coordinates lifted from Y_{2}, the y’s coordinates lifted from the fibre of ϕ_{2} in Y_{1} through p and the z’s are coordinates on the fibre of ϕ_{1} through p. A smooth vector field on X is an element of V_{dw}(X) if and only if expressed in terms of these coordinates it is a smooth linear combination of the basis vector fields

x^{2} \frac{∂}{∂x}, x^{2} \frac{∂}{∂t}, x\frac{∂}{∂y} and \frac{∂}{∂z} .

Thus the theorem just states that this Lie algebra of vector fields has a resolution in the sense described above. Thus the spaces X_{dw}^{2} and X_{dw}^{3} and appropriate maps must be constructed. These spaces will be constructed from X^{2} and X^{3} respectively by iterated radial blow up.

Let us consider first X_{dw}^{2}; for simplicity assume that ∂X is connected and that the fibrations have connected fibres. The first stage in its construction is the blow up of the fibre diagonal at the corner of X^{2} corresponding to the total fibration ψ. In the corner (∂X)^{2} ⊂ X^{2} of X^{2} consider the submanifold which is the ψ-fibre diagonal, namely

B_{2} = {(p, q) ∈ (∂X)^{2}; ψ(p) = ψ(q)}.

This is a closed embedded submanifold of the corner (which is a manifold without boundary); in terms of local coordinates as discussed above, x, y, t, z and x′, y′, t′, z′ in the two factors of X it is y = y′, x = x′ = 0. As such it is a p-submanifold of X^{2} so it can be blown up radially to give the new manifold with corners [X^{2}; B_{2}] and associated blow-down map β : [X^{2}; B_{2}] ⟶ X^{2}.

The second stage of the blow up involves the ‘lift’ of the fibre diagonal associated to ϕ̃_{1}. The fibre diagonal, defined near the boundary,

B′_{1} = {(p, q); ϕ̃_{1}(p) = ϕ̃_{1}(q)}

lifts under the first blow up to a p-submanifold near the boundary and B_{1} is by definition the intersection of this lift with the boundary. It is again a boundary p-submanifold. Then the stretched double space is given by iterated blow up

X_{dw}^{2} = [X^{2}; B_{2}; B_{1}].

The total blow down map is therefore a generalized blow down map. The conditions in the first four conditions can also be checked.

The construction of X_{dw}^{3} is similar but more involved since it has four stages. In X^{3} there are three analogues of B_{2}, namely the corresponding fibre diagonals in the three pairs of factors; these can be denoted B_{2,O}, O = S, C, F. These intersect in the triple fibre diagonal B_{2,T} . All are boundary p-submanifolds. The first step is to blow up B_{2,T} . Under this blow up the three fibre diagonals lift to disjoint p-submanifolds which can therefore be blown up independently.

The submanifold B_{1} similarly has three analogues denote, B_{1,O}, for O = S, C, F, in the blown up version of X^{3} produced so far. They are all p-submanifolds intersecting in pairs in the same p-submanifold B_{1,T} . The remaining two stages in the construction of X_{dw}^{3} are the blow up of B_{1,T} after which the remaining B_{1,O} lift to be disjoint and are then blown up. This defines X_{dw}^{3}.

The main step in checking the properties required above of a resolution of the Lie algebra V_{dw}(X) is to show the existence of the stretched projections π_{O,dw}, for O = S, C, F. This is done using the commutation properties of blow up outlined above as follows for O = F with B̃_{1} = B_{1,C} ∪ B_{1,S} and B̃_{2} = B_{2,C} ∪ B_{2,S} :

X_{dw}^{3} = [X^{3}; B_{2,T}; B_{2,F}; B̃_{2}; B_{1,T}; B_{1,F}; B̃_{1}]

⟶^{β(B̃_{1})} [X^{3}; B_{2,T}; B_{2,F}; B̃_{2}; B_{1,T}; B_{1,F}],

≡ [X^{3}; B_{2,F}; B_{2,T}; B̃_{2}; B_{1,F}; B_{1,T}], B_{2,T} ⊂ B_{2,F} , B_{1,T} ⊂ B_{1,F} ,

⟶^{β(B_{1,T})} [X^{3}; B_{2,F}; B_{2,T}; B̃_{2}; B_{1,F}]

≡ [X^{3}; B_{2,F}; B_{2,T}; B_{1,F}; B̃_{2}], B_{1,F} ⊂ B̃_{2}

⟶^{β(B̃_{2})} [X^{3}; B_{2,F}; B_{2,T}; B_{1,F}] ≡ [X^{3}; B_{2,F}; B_{1,F}; B_{2,T}]], B_{2,T} ⊂ B_{1,F}

⟶^{β(B_{2,T})} [X^{3}; B_{2,F}; B_{1,F}] ≡ [X × X^{2}; X × B_{2}; X × B_{1}] ≡ X × X_{dw}^{2}

⟶^{π} X_{dw}^{2}.

It is then relatively straightforward to check the desired properties of these maps.

The calculus of operators given by the results outlined in the previous section then gives a description of the resolvent.

Proposition. The Laplacian of the metric g in (1) is an element of Diff_{dw}^{2}(X) for the corresponding doubly-warped structure and the resolvent (Δ − σ)^{−1} ∈ Ψ_{dw}^{−2}(X) is a holomorphic function of σ ∈ ℂ \ [0, ∞).

Proof. This result can be shown constructively, essentially as the same result is obtained in the boundaryless case using the normal operator and the usual symbol map.

§3: Double zero metrics

Next I shall briefly describe the proof of the existence of generalized eigenfunctions as in (2), restricting the discussion to the special case of a double-zero metric, where the two fibrations maps ϕ_{i} are isomorphisms and can be forgotten. This represents well the changes needed to the proof of the corresponding fact in the scattering calculus ([6]) and the fibred-cusp calculus ([8], [3]).

In terms of the coordinates introduced above there are no z or y variables; the metric becomes

g = \frac{h}{x^{4}}

where h is a non-degenerate metric on X and x is a boundary defining function equal to the h-distance to second order.

In the special case that x is the distance for h :

(3) Δ = −(x^{2}∂_{x})^{2} + (2n − 4)x^{3}∂_{x} + x^{4}Δ_{h(x)} + x^{2}W, W ∈ V_{dz}(X)

where Δ_{h(x)} is the tangential Laplacian of the x-dependent metric. In the general case the same decomposition is valid up to a term in x^{2} Diff_{dz}^{2}(X) (which annihilates 1). An easy inductive argument then shows that there are formal solutions as in (2). More precisely for any f ∈ C^{∞}(∂X) there is a function f̃ ∈ C^{∞}(X), unique up to Ċ^{∞}(X), i.e. in Taylor series at the boundary, such that

(Δ − λ^{2})ũ_{f} = e_{f} ∈ Ċ^{∞}(X), ũ_{f} = e^{iλ/x}x^{n−1}f̃, 0 ≠ λ ∈ ℝ, n = dim X.

The actual eigenfunction in (2) is then constructed as

(4)  u_{f} = ũ_{f} − lim_{ϵ↓0}(Δ − (λ − iϵ)^{2})^{−1}e_{f}.

Even if there is discrete spectrum this limit exists.

Thus the proof of the existence of u_{f} reduces to an appropriate form of the limiting absorption principle, the examination of the limit of the resolvent on the real axis. This is not really an ‘elliptic’ problem but rather has more similarity to hyperbolic equations. To explain this I shall first discuss some properties of the calculus Ψ_{dz}^{∗}(X) which is defined as a special case of the discussion above.

The double-zero calculus has a global quantization map. Consider the vector bundle ^{dz}T^{∗}X which is the dual of the ‘structure bundle’ dzT X of which V_{dz}(X) = C^{∞}(X;^{dz}T X) is the space of all sections. Radial compactification of the fibres of this bundle defines a compact manifold with corners which I shall denote^{dz}\overline{T}^{∗}X. Since X is a manifold with boundary (assumed connected for notational simplicity) ^{dz}\overline{T}^{∗}X has two boundary hypersurfaces (assuming that dim X > 1 as we shall). Let ρ_{∂} and ρ_{σ} be defining functions for these two boundary hypersurfaces. Then for any integers k, l the spaces ρ_{∂}^{k}ρ_{σ}^{l}C^{∞}(^{dz}\overline{T}^{∗}X) are special case of the polyhomogeneous conormal distributions discussed above. There is then a global quantization map for all k, l which is itself independent of k, l

q : ρ_{∂}^{k}ρ_{σ}^{−l}C^{∞}(^{dz}\overline{T}^{∗}X) ⟶ ρ^{k}Ψ_{dz}^{l}(X).

The map q has null space contained in Ċ^{∞}(^{dz}\overline{T}^{∗}X) and range a complement to ρ^{∞}Ψ_{dz}^{−∞}(X). In consequence it induces a ‘full symbol isomorphism’

q̃: ρ_{∂}^{k}ρ_{σ}^{−l}C^{∞}(^{dz}\overline{T}^{∗}X)/Ċ^{∞}(^{dz}\overline{T}^{∗}X) ⟶ ρ^{k}Ψ_{dz}^{l}(X)/ρ^{∞}Ψ_{dz}^{−∞}(X)

and thus induces a ‘star product’ on the domain space, the space of Laurent series at the boundary, i.e. an associative product given to any order by bilinear differential operators

a ⋆ b ≡ ∑_{j=0}^{∞} P_{j}(a, b), P_{0}(a, b) = ab.

The microlocality of this product arises here from the limiting commutativity of the Lie algebra W. The principal symbol map, defined as the leading part of the inverse of q, is independent of all choices and gives a short exact sequence for each k, l

0 ⟶ ρ_{∂}^{k+1}Ψ_{dz}^{l−1}(X) ↪ ρ_{∂}^{k}Ψ_{dz}^{l}(X) ⟶^{σ_{l,k}}

ρ_{∂}^{k}ρ^{−l}C^{∞}(^{dz}\overline{T}^{∗}X)/ρ_{∂}^{k+1}ρ_{σ}^{−l+1}C^{∞}(^{dz}\overline{T}^{∗}X) ⟶ 0.

This identification (even though it is by no means unique) shows shows that there is an associated notion of wavefront set. Namely for an element of the algebra the wavefront set is the support (in the boundary) of the associated Laurent series:

WF′_{dz}(A) = supp([a]) ⊂ C_{dz}(X) = ∂^{dz}\overline{T}^{∗}X,
A ∈ ρ^{k}Ψ_{dz}^{l}(X), q̃([a]) = [A].

This wavefront set makes the calculus microlocal in the sense that, WF′_{dz}(A) ⊂ WF′_{dz}(A)∩WF′_{dz}(B). Then for extendible distributions on X one can set, as in the boundaryless case,

WF_{dz}(u) = ⋂ {WF′_{dz}(A); A ∈ ρ^{k}Ψ_{dz}^{l}(X), Au ∈ Ċ^{∞}(X)}

with the wavefront set independent of k, l.

This wavefront set has the usual properties, for example

u ∈ C^{−∞}(X), WF_{dz}(u) = ∅ ⟹ u ∈ Ċ^{∞}(X)

and if the elliptic set Ell_{k,l}(A) of A ∈ ρ^{k}Ψ_{dz}^{l}(X) is defined to be the set where σ_{k,l}(A) is invertible then

u ∈ C^{−∞}(X), Au ∈ Ċ^{∞}(X) ⟶ WF_{dz}(u) ∩ Ell_{k,l}(A) = ∅.

Returning to the question of eigenfunctions, suppose u ∈ C^{−∞}(X) is a tempered eigenfunction of the Laplacian, so

(Δ − λ^{2})u = 0.

Then it follows immediately that

WF_{dz}(u) ⊂ {σ_{0,2}(Δ) = λ^{2}} ⊂ C_{dz}X.

The symbol of the Laplacian is represented in ρ_{σ}^{−2}C^{∞}(^{dz}\overline{T}^{∗}X) by the square of the metric length function on ^{dz}T^{∗}X. Thus the wavefront set of an eigenfunction u must be contained in the characteristic set of Δ − λ^{2}, which is the appropriate sphere bundle over the boundary

WF_{dz}(u) ⊂ {|Ξ|^{2} = λ^{2}} ⊂ ^{dz}T_{∂X}^{∗}X ⟶ C_{dz}X.

To analyze the limit in (4) a uniform (in ϵ) version of this wavefront set, of finite order relative to weighted Sobolev spaces, can be used. Thus, if u_{ϵ} ∈ C^{−∞}(X) is a family of distributions depending on ϵ ∈ P, a parameter space, then a point Z ∈ C_{dz}X is not in the uniform wavefront set WF_{dz,P}^{k,l} relative to the finite order Sobolev space H_{dz}^{k,l}(X) if it is bounded in some such Sobolev space and there is an operator A ∈ ρ^{−l}Ψ_{dz}^{k}(X) which is elliptic at Z and such that Au_{ϵ} is bounded in L_{dz}^{2}(X).

The starting point is the self-adjointness of Δ from which it follows that u_{ϵ}, the solution of (4), i.e.

(5)  (Δ − (λ − iϵ)^{2})u_{ϵ} = g ∈ Ċ^{∞}(X), ϵ > 0, 0 ≠ λ ∈ ℝ,

has ϵu_{ϵ} bounded in L_{dz}^{2}(X) for ϵ ∈ (0, 1].

Consider a family v_{ϵ} which is bounded in H_{dz}^{k′,l′}(X) for ϵ ∈ (0, 1] = P and some k′, l′ and suppose that

(6)  (Δ − (λ − iϵ)^{2})v_{ϵ} = f_{ϵ} is bounded in Ċ^{∞}(X).

Directly from the definition of the uniform wavefront set it follows that

WF_{dz,P}^{k,l} (v_{ϵ}) ⊂ {|Ξ| = λ^{2}} ∀ k, l, P = (0, 1].

Using commutation arguments as in [6] improved microlocal bounds can also be deduced from (6). First there are no singularities in the ‘incoming’ subset.

Proposition. For a family v_{ϵ} bounded in H_{dz}^{k′,l′}(X) for some k′, l′ satisfying (6) and any k, l ∈ ℝ

WF_{dz,P}^{k,l} (v_{ϵ}) ∩ \{−λ\frac{dx}{x^{2}}\} = ∅.

Secondly Hörmander’s theorem on the propagation of singularities is valid in this context for the ‘Hamilton vector field’ of the Laplacian Proposition. For a family v_{ϵ} bounded in H_{dz}^{k′,l′}(X) for some k′, l′ satisfying (6) and any k, l ∈ ℝ, WF_{dz,P}^{k,l} (v_{ϵ}) is a union of maximally extended curves in {τ^{2} + |ζ|^{2} = λ^{2} of the vector field τζ · ∂_{ζ} − |ζ|^{2}∂_{τ} .

Combining these two proposition it follows that WF_{dz,P}^{k,l} (v_{ϵ}) is confined to the ‘outgoing’ surface λ\frac{dx}{x^{2}}. There its order can be bounded,

Proposition. For a family v_{ϵ} bounded in H_{dz}^{k′,l′}(X) for some k′, l′ satisfying (6) and any k, l ∈ ℝ WF_{dz,P}^{k,l} (v_{ϵ})

WF_{dz,P}^{k,l} (v_{ϵ}) = ∅ ∀ k and l < −\frac{1}{2}.

Now applying these results to v_{ϵ} = ϵu_{ϵ} it follows that ϵu_{ϵ} is bounded in H_{dz}^{k,l}(X) for ϵ ∈ P, any k and any l < −\frac{1}{2}. It is therefore a precompact family in the same spaces. In fact this argument applies uniformly in λ near any given non-zero λ′ ∈ ℝ. Thus, if ∥ϵu_{ϵ}∥_{k′,l′}, l′ < −\frac{1}{2}, does not tend to zero as ϵ ↓ 0 then along some subsequence it converges to a non-zero limit which is an eigenfunction and has WF_{dz} contained in the outgoing set.

Proposition. If 0 ≠ λ ∈ ℝ and u ∈ C^{−∞}(X) satisfies

(Δ − λ^{2})u = f ∈ Ċ^{∞}(X), WF_{dz}(u) ∈ \{λ\frac{dx}{x^{2}}\}

then

(7)  u = e^{iλ/x}x^{n−1}u′, u′ ∈ C^{∞}(X).

This explicit form of the function allows a version of Green’s formula to be proved.

Lemma. If u is as in the proposition above then

lim_{δ↓0} ∫_{x<δ} \overline{u}(Δ − λ^{2})udg = ∫_{∂X} |u′|^{2}dh.

As a corollary, for an actual eigenfunctions u which is outgoing it follows that u′ = 0 on ∂X and then by a modified form of the propositions above that u ∈ Ċ^{∞}(X). In fact the proof shows that for λ ∈ [δ, 1/δ], δ > 0, the span of such eigenfunctions forms a subspace of Ċ^{∞}(X) with bounded sphere, it is therefore finite dimensional.

Proposition. For the Laplacian of a double-zero metric the embedded L^{2} spectrum is discrete and of finite multiplicity.

Returning now to the family u_{ϵ} satisfying (5) we can simply project it onto the L^{2} eigenspace with eigenvalue λ^{2} if this is zero

u_{ϵ} = u′_{ϵ} + u′′_{ϵ},

where u′′_{ϵ} is the part in the eigenspace. Necessarily, ϵu′′_{ϵ} is bounded in Ċ^{∞}(X) for ϵ ∈ (0, 1]. Applying the operator it follows that u′_{ϵ} satisfies essentially the same condition in that

(Δ − (λ − iϵ)^{2})u′_{ϵ} = g_{ϵ} is bounded in Ċ^{∞}(X).

Repeating the same arguments as before but using the orthogonality to the eigenfunctions (if there are any) it follows that u′_{ϵ} ⟶ u is of the form (7). This finally proves the existence of the generalized eigenfunction as in (2) in this case.

§4: Poisson operator and scattering matrix

Construction of the Poisson operator P_{λ} amounts to the solution of (2) for f a delta function at a general boundary point, ȳ. This can be done for a double-zero metric much in terms of Legendre distributions, much as in [10] for scattering metrics. In the present case the construction of a parametrix for P_{λ} is local, it takes place in a small neighbourhood of the point ȳ.

Take local coordinates near ȳ in which ȳ = 0 and begin with the ‘ansatz’

v = (2π)^{−n+1} ∫ exp(i\frac{y · u}{x^{2}} − i\frac{α(u)}{x})x^{l}a(x, y, u)du.

Here α(u) is a smooth function of u ∈ ℝ^{n−1} to be chosen and similarly a(x, y, u) is a smooth function in x ≥ 0 with support in |u| < \frac{1}{2}, |y−ȳ| < δ. Thus the integral certainly converges for x > 0. Integration against a ‘test function’ ϕ ∈ C^{∞}(ℝ^{n−1}) and use of stationary phase shows that

\begin{matrix}∫ v(x, y)ϕ(y)dy = x^{l}(2π)^{−n+1} ∫ exp(i\frac{y · u}{x^{2}} − i\frac{α(u)}{x})a(x, y, u)dudy \\ = x^{l+2(n−1)}e^{−iα(0)/x}c(x), c ∈ C^{∞}([0, 1)), c(0) = a(0, 0, 0)ϕ(0). \end{matrix}

Thus the ‘initial conditions’ to give the correct incoming coefficient are

l = −(n − 1), α(0) = λ, a(0, 0, 0) = 1.

Moreover, using (3) in applying the Laplacian to v and using integration by parts shows that (Δ − λ^{2})v = g is of the same form. If the coordinates are chosen to be Riemmanian normal coordinates in the boundary and to first order off the boundary then

(8) \begin{matrix}g = (2π)^{−n+1} ∫ exp(i\frac{y · u}{x^{2}} − i\frac{α(u)}{x})x^{l}b(x, y, u)du, \\ b(x, y, u) = ((α + 2u · ∂_{u}α)^{2} + |u|^{2} − λ^{2})a + x^{2}b′, b′ C^{∞}. \end{matrix}

Thus the ‘eikonal equation’ for α is radial in the u variables and can be written in polar coordinates in the form

(r^{− \frac{1}{2}} α(r))′ = \frac{(λ^{2} − r^{2})^{\frac{1}{2}}}{2r^{\frac{3}{2}}} .

This has a unique smooth solution in |u| < λ with α(0) = λ.

Choosing α to be this function removes the leading term in (8). The next term, vanishing quadratically with respect to the first, is given by the transport equation and can be made to vanish by demanding that

u · ∂_{u}a(0, 0, u) = 0 near u = 0.

Here the constant term has been removed by the choice of l = −(n−1). This also has a unique smooth solution with a(0, 0, 0) = 1. The presence of constant terms means that lower order transport equations have unique solutions which are smooth at u = 0. Thus, at an essentially formal level it can be arranged that b in (8) vanishes to all orders at x = 0.

There is an obstruction to continuing the solution in this form, in the singularity in the phase α which occurs at |u| = λ. Thus in practice it is necessary to insert a cut off function in a to restrict the support to |u| ≤ R < λ. This leads to an error term (Δ − λ^{2})v = g with g of the form (8) for b = x^{2}b′′ and b′′ supported in an annular region R − δ < |u| < R + δ where δ can be chosen appropriately small.

This failure amounts to the appearance of a ‘caustic’ for the Legendre manifold involved. Rather than deal directly with the invariance properties of oscillatory integrals of the form in the ansatz it is convenient to use the locality of the construction (so far) and to blow up the base point. Thus, instead of considering the solution on X it will be lifted to

[X, {ȳ}]

which is just X with the initial point blown up. In the local coordinates introduced above, in which ȳ = 0, this is simply the introduction of polar coordinates in the variables (x, y),

(x, y) = ρ(ω_{0}, ω′), ω ∈ 𝕊^{n−1}, ω_{0} ≥ 0.

Thus the blown up space is a compact manifold with corner of codimension two. However, the initial parametrix is rapidly decreasing near the ‘old’ boundary (defined by ω_{0} = 0) and so attention can be limited to a neighbourhood of the interior of the new face {ρ = 0} introduced by the blow up. In this region the simpler projective coordinates

x, Y = \frac{y}{x}

are valid.

A straightforward computation shows that in these (singular) coordinates the ansatz above gives a Legendre distribution in the sense of [10] and the Laplacian becomes an elliptic scattering differential operator in the sense of [6]. The metric of which it is the Laplacian is not quite a ‘scattering metric’ in the sense of [6] however, since it does not have the splitting property demanded there. It is still a smooth nondegenerate metric on the scattering tangent bundle. As a result the construction of the Poisson operator given in [10] can be mimicked to give the Poisson operator here, as a Legendre distribution. The structural difference in the metric leads to a difference in the behaviour of the geodesic flow and hence the Legendre manifold. Namely all the rescaled bicharacteristics starting from the point Y = 0 on the incoming radial set have end point at Y = 0 on the outgoing radial set. It is not the case that the flow remains in the fibre above Y = 0, as it does in the original space, rather there is a natural focussing effect.

The construction of the Poisson operator as a Legendre distribution and the geometry of this Legendre manifold leads directly to the conclusion that the scattering matrix is a pseudodifferential operator, just as in [10] it is seen to be a Fourier integral operator.

References

1. C.L. Epstein and R.B. Melrose and G. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1–106.

2. L. Hörmander, Fourier integral operators, I, Acta Math. 127, 79–183.

3. R. Mazzeo and R.B. Melrose, Pseudodifferential operators on manifolds with fibred boundaries, In preparation.

4. R.B. Melrose, Calculus of conormal distributions on manifolds with corners, Int. Math. Res. Notes 21 (1992), 67-77.

5. , The Atiyah-Patodi-Singer index theorem, A K Peters, Wellesley, Mass, 1993.

6. , Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Spectral and scattering theory (M. Ikawa, ed.), Marcel Dekker, 1994.

7. , Geometric Scattering, Cambridge University Press, 1995.

8. , Fibrations, compactifications and algebras of pseudodifferential operators, Partial Differential Equations and Mathematical Physics. The Danish-Swedish Analysis Seminar, 1995 (Lars Hörmander and Anders Melin, ed.), Birkhäuser, 1996, pp. 246–261.

9. , Differential analysis on manifolds with corners, In preparation.

10. R.B. Melrose and M. Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1996), 389–436.