GENERALIZED BACKSCATTERING AND THE LAX-PHILLIPS TRANSFORM

RICHARD MELROSE AND GUNTHER UHLMANN

Dedicated to Vesselin Petkov on the occasion of his 65th birthday

Abstract. Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle ω = Sθ in terms of the incoming angle with S orthogonal and Id −S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials.

Contents

Introduction 1 1. Lax Phillips transform 2 2. Sobolev bounds 5 3. Generalized backscattering transform 6 4. Proof of the main result 9 5. Fredholm property 12 References 12

Introduction

The inverse scattering problem in the two body case consists of determining a potential V by measuring the scattering amplitude a_{V} (λ, ω, θ) where λ denotes the frequency of an incoming plane wave with direction ω and θ denotes the outgoing direction. This is an overdetermined problem except in dimension one. In this note we consider determined problems where the set of possible angles θ and ω is restricted to an n − 1 dimensional subset of the complement of the diagonal in the product of the sphere with itself. We use the time dependent approach to scattering of Lax-Phillips [LP], [P]. This is a based on the classical wave equation rather than the time dependent or stationary Schrödinger equation and therefore allows the properties of the wave equation, especially the finite speed of propagation of the solutions and the precise description on the propagation of singularities, to be effectively exploited. In particular the Lax-Phillips modified Radon transform (their free-space translation-representation), reduces the n-dimensional problem to a one dimensional problem with lower order term arising from the potential.

If S is an n-dimensional orthogonal transformation such that Id −S is invertible, then the modified backscattering transform determined by S, for a potential V, is obtained by composing the restriction of the scattering kernel κ_{V} (s, ω, θ) (the inverse Fourier-Laplace transform in λ of the scattering amplitude) to ω = Sθ with a linear map L_{S} (the generalized inverse of the linearization of the map at V = 0). In the Main Theorem in Section 3, it is shown that if Ḣ^{\frac{n+1}{2}} (B(ρ)) is the Sobolev space of functions with support in the closed ball of radius ρ then

(1)  Ḣ^{\frac{n+1}{2}} (B(ρ)) ∋ V ⟼ L_{S}(κ_{V} (s, Sθ, θ)) ∈ Ḣ^{\frac{n+1}{2}} (B(ρ))

is an entire and globally Fredholm non-linear map of index zero. Indeed this map is a local isomorphism near potentials forming an open set with complement of codimension at least two (see Proposition 2 in Section 5).

Related results, in a slightly different setting for the true backscattering, S = − Id but including two dimensions and non-compact supports, have been obtained by Eskin and Ralston [ER1, ER2, ER3]. A different method to prove generic uniqueness was given in [S] in dimension 3 for compactly supported potentials. The use of hyperbolic equations for the inverse backscattering problem has also been considered in several papers; see for instance [BLM], [GU], [Mo], [StU]. The lecture notes [MU] contain most of what we do here. In [Mel], [U], we gave a sketch of the proof of the main Theorem here for the case S = − Id. The case of even dimensions n > 2, also for S = − Id, using similar methods to [MU], [Mel] and [U], was considered in [W]. Melin has developed an alternative approach to the inverse backscattering problem using the ultrahyperbolic equation [Me1], [Me2].

We leave open the question of whether a map such as (1) is a global isomorphism, or a local isomorphism near each potential. The problem of determining partial information of the potential, especially its singularities, from backscattering or other (formally) determined information has been considered in the papers [GU], [J], [OPS], [SuU], [R], [RV] and in the recent preprint [BM].

The authors thank the anonymous referee and Jeff Lagarias for comments on the manuscript.

1. Lax Phillips transform

We briefly recall the approach of Lax and Phillips to scattering theory in odddimensional Euclidean space. Since it suffices for the present problem we give a simplified formulation of their theory.

The Lax Phillips theory is founded on the Radon transform:

(2)  \begin{matrix}Rf(s, ω) = ∫_{HS(s,ω)} f(x)dH_{x}, \\ R : C_{c}^{∞}(ℝ^{n}) ⟶ C_{c}^{∞}(ℝ × 𝕊^{n−1}) \end{matrix}

where H_{x} is surface measure on HS(s, ω) = {x · ω = s}. The formal transpose, R^{t}, of R is given by

(3)  \begin{matrix}R^{t} : C_{c}^{∞}(ℝ × 𝕊^{n−1}) ⟶ C^{∞}(ℝ^{n}), \\ R^{t}g(x) = ∫_{𝕊^{n−1}} g(x · ω, ω)dω. \end{matrix}

Of particular importance here is the fact that the Radon transform intertwines the n-dimensional and the one-dimensional Laplacians (for any n ≥ 2)

(4)  RΔf = D_{s}^{2}Rf ∀ f ∈ C_{c}^{∞}(ℝ^{n})

where Δ is the positive Laplacian and D_{s} = \frac{1}{i}∂_{s}. Moreover there is an inversion and a Plancherel formula; for any f, g ∈ C_{c}^{∞}(ℝ^{n})

(5) \begin{matrix}f = \frac{1}{2(2π)^{n−1}} R^{t}(|D_{s}|^{n−1}Rf), \\ ∫_{ℝ^{n}} f(x)\overline{g(x)}dx = \frac{1}{2} \frac{1}{(2π)^{n−1}} ∫_{ℝ×𝕊^{n−1}} (|D_{s}|^{\frac{n−1}{2}} Rf)(s, ω)\overline{(|D_{s}|^{\frac{n−1}{2}} Rg)(s, ω)}dsdω. \end{matrix}

The range of R on C_{c}^{∞}(ℝ^{n}) was characterized by Helgason in [Hel]. Its closure

in an appropriate topology is simpler. Thus if, n ≥ 3 is odd the operator Ds^{\frac{n−1}{2}} · R extends by continuity to an isometric isomorphism

(6) R_{n} = Ds^{\frac{n−1}{2}} ·R : L^{2}(ℝ^{n}) ⟶ \{k ∈ L^{2}(ℝ×𝕊^{n−1}); g(−s, −ω) = (−1)^{\frac{n−1}{2}} g(s, ω)\}

and R^{t} · Ds^{\frac{n−1}{2}} extends by continuity to be its inverse.

The modified Radon transform of Lax and Phillips is defined to be

(7)  LP \binom{u_{0}}{u_{1}} = 2^{\frac{1}{2}} (2π)^{\frac{n−1}{2}} \{Ds^{\frac{n+1}{2}} (Ru_{0})(s, ω) − Ds^{\frac{n−1}{2}} (Ru_{1})(s, ω)\} .

For n ≥ 3 odd it is an injective map

(8)  LP : C_{c}^{∞}(ℝ^{n}) × C_{c}^{∞}(ℝ^{n}) ⟶ C_{c}^{∞}(ℝ × 𝕊^{n−1})

which intertwines the free wave group and the translation group:

(9) \begin{matrix}LP ·U_{0}(t) = T_{t} · LP, T_{t}v(s, ω) = v(s − t, ω), \\ U_{0}(t)\binom{u_{0}}{u_{1}} = \binom{u(t)}{D_{t}u(t)}, (D_{t}^{2} − Δ)u(t) = 0, u(0) = u_{0}, D_{t}u(0) = u_{1}. \end{matrix}

In particular, if u is the solution of the Cauchy problem for the wave equation as in (9) and

(10)  k(t, s, ω) = LP · U_{0}(t)\binom{u_{0}}{u_{1}} ∈ C^{∞}(ℝ × ℝ × 𝕊^{n−1})

then k(t, s, ω) = k_{0}(s − t, ω) is a solution of the first order differential equation

(11)  (D_{t} + D_{s})k(t, s, ω) = 0 in ℝ × ℝ × 𝕊^{n−1}.

This is in essence the free-space translation representation of Lax and Phillips. Rather than adopting their approach of constructing a perturbed translation representation for the wave equation with potential we use the same ‘free’ Lax Phillips transform and observe its effect on the solution to the perturbed Cauchy problem

(12)  \begin{matrix}U_{V} (t)\binom{u_{0}}{u_{1}} = \binom{u(t)}{D_{t}u(t)}, \\ (D_{t}^{2} − Δ − V (x))u(t) = 0, u(0) = u_{0}, D_{t}u(0) = u_{1}, \end{matrix}

where V ∈ C_{c}^{∞}(ℝ^{n}). Namely if

(13)  k_{V} (s, t, ω) = LP · U_{V} (t)\binom{u_{0}}{u_{1}} ∈ C^{∞}(ℝ × ℝ × 𝕊^{n−1}),

then

(14)  \begin{matrix}(D_{t} + D_{s})k_{V} (t, s, ω) \\ = 2^{\frac{1}{2}} (2π)^{\frac{n−1}{2}} \{−Ds^{\frac{n−1}{2}} RD_{t}^{2}u(t, ·) + Ds^{\frac{n+3}{2}} Ru(t, ·)\} \\ = −2^{\frac{1}{2}} (2π)^{\frac{n−1}{2}} Ds^{\frac{n−1}{2}} R[V (·)u(t, ·)]. \end{matrix}

Using the inversion formula it follows that

(15)  (D_{t} + D_{s})k_{V} (t, s, ω) + V_{LP}k_{V} (t, s, ω) = 0

where V_{LP} is an operator on C^{∞}(ℝ × 𝕊^{n−1}) :

(16)  V_{LP} = \frac{1}{2(2π)^{n−1}} Ds^{\frac{n−1}{2}} · R · V · R^{t}Ds^{\frac{n−3}{2}} .

Thus if supp(V) ⊂ {|x| ≤ ρ} the operator V_{LP} defined by (16) has Schwartz kernel V_{LP}(s, ω, s′, ω′) supported in the region

(17)  supp(V_{LP}) ⊂ {(s, ω′, s′, ω) ∈ ℝ × 𝕊^{n−1} × ℝ × 𝕊^{n−1}; |s|, |s′| ≤ ρ} .

There is a unique fundamental solution, which is to say a distribution satisfying

(18)  \begin{matrix}(D_{t} + D_{s} + V_{LP})E_{LP}(t, s, ω; s′, θ) = 0 \\ E_{LP}(0, s, ω; s′, θ) = δ(s − s′)δ_{θ}(ω). \end{matrix}

Standard properties of the wave equation imply that

(19)

singsupp(E_{LP}) ⊂
{s′ − s + t = 0, θ = ω} ∪ {s′ + s + t = 0, θ = −ω, |s| ≤ ρ, |s′| ≤ ρ}.

¿From this it follows that the continuation problem can also be solved, so for each θ ∈ 𝕊^{n−1} there is a unique distribution

(20)  α(t, s, ω, θ) ∈ C^{−∞}(ℝ × ℝ × 𝕊^{n−1} × 𝕊^{n−1}),

satisfying

(21) (D_{t} + D_{s})α + V_{LP}α = 0 in ℝ × ℝ × 𝕊^{n−1} and α(t, s, ω; θ) = δ(s − t)δ_{θ}(ω) in t < −ρ

where ρ = sup{|x|; x ∈ supp(V)}.

It follows that

(22)  α(t, s, ω; θ) = κ_{V} (t − s, ω; θ) in s > ρ

where κ_{V} ∈ C^{−∞}(ℝ × 𝕊^{n−1} × 𝕊^{n−1}) is the scattering kernel. Here one can think of α as the free wave

(23)  α_{0}(t, s, ω; θ) = δ(s − t)δ_{θ}(ω)

propagating in from the left and striking the ‘potential’ which is confined to the region |s| ≤ ρ. Once it has passed through the potential it again freely propagates to the right. Thus the kernel κ_{V} (t, ω; θ) represents the end result of the interaction.

The scattering amplitude in the ordinary sense is the Fourier-Laplace transform of κ_{V} continued to the real axis. We define the generalized backscattering transform below directly from κ_{V} .

2. Sobolev bounds

We will consider potentials V with fixed support and finite Sobolev regularity. So, for ρ ∈ (0, ∞), set

(24) Ḣ^{\frac{n+1}{2}} (B(ρ)) = \{V ∈ L^{2}(ℝ^{n}); V (x) = 0 in |x| > ρ,

D^{α}V ∈ L^{2} ∀ |α| ≤ \frac{n + 1}{2}\}.

The choice of Sobolev order here is not critical; it is convenient that \frac{n+1}{2} is an integer and rather more important that \frac{n+1}{2} > \frac{n}{2}. The latter condition means that Ḣ^{\frac{n+1}{2}} B(ρ)) is an algebra. In fact the usual Sobolev spaces are then modules over these for an appropriate range of orders.

Lemma 1. (Gagliardo-Nirenberg, see [Ad]) For any k ∈ ℕ with k > n/2 and any s ∈ ℝ satisfying −k ≤ s ≤ k

(25)  H^{k}(ℝ^{n}) · H^{s}(ℝ^{n}) ⊂ H^{s}(ℝ^{n}).

In particular, if s ∈ ℝ and −\frac{n+1}{2} ≤ s ≤ \frac{n+1}{2}, then

(26)  Ḣ^{\frac{n+1}{2}} (B(ρ)) · H^{s}(ℝ^{n}) ⊂ Ḣ^{s}(B(ρ)).

Lemma 2. For any k (∈ ℤ for simplicity) the normalized Radon transform in (6) gives a bounded map

(27)  \begin{matrix}R_{n} : Ḣ^{k}(B(ρ)) ⟶ Ḣ^{k}([−ρ, ρ] × 𝕊^{n−1}) = \{u ∈ & H^{k}(ℝ × 𝕊^{n−1}); \\ u(s, θ) = 0 in |s| > ρ\}. \end{matrix}

Proof. For k = 0, this is (6) which is a consequence of the L^{2} boundedness of the Fourier transform. Consider the case k > 0. We know that R (and hence R_{n}) intertwines Δ with D_{s}^{2}. Thus if f ∈ C_{c}^{∞}(ℝ^{n}) then

(28)  D_{s}^{2}R_{n}f = R_{n}Δf.

Since R_{n} is a partial isometry on L^{2},

(29)  ⟨R_{n}f, D_{s}^{2}R_{n}f⟩_{L^{2}} = ⟨Δf, f⟩.

By continuity then, f ∈ Ḣ^{1}(B(ρ)) ⟹ D_{s}R_{n}f ∈ L^{2}. Repeating this argument a finite number of times shows that

(30)  f ∈ Ḣ^{k}(B(ρ)) ⟹ D_{s}^{j}R_{n}f ∈ L^{2}([−ρ, ρ] × 𝕊^{n−1}) 0 ≤ j ≤ k.

To get tangential regularity, suppose that W is a C^{∞} vector field on the sphere. Then

(31)  \begin{matrix}W R_{n}f(s, θ) = c_{n}Ds^{\frac{n−1}{2}} W ∫ δ(s − x · θ)f(x)dx \\ = ∑_{j=1}^{n} q_{j}(θ)D_{s}R_{n}(x_{j}f), W (x · θ) = ∑_{j=1}^{n} x_{j}q_{j}(θ). \end{matrix}

Thus W R_{n}f ∈ L^{2}. Repeating this argument we conclude that (27) holds for k ≥ 0. The same type of argument applies to R_{n}^{t}. Thus

(32)  R_{n}^{t}u(x) = c_{n} ∫_{𝕊^{n−1}} δ(s − x · ω)Ds^{\frac{n−1}{2}} u(s, ω)ds

is bounded from L^{2}([−ρ, ρ] × 𝕊^{n−1}) into L^{2}(B(ρ)). Direct differentiation therefore shows that it is bounded from H^{k}([−ρ, ρ] × 𝕊^{n−1}) into H^{k}(B(ρ)) for k ∈ ℕ. By duality it follows that (27) holds for k ∈ −ℕ, and hence for all k ∈ ℤ as claimed. □

Note that, from the proof above,

R^{t} : {u ∈ C^{−∞}(ℝ × 𝕊^{n−1});D_{s}^{j}u ∈ L_{loc}^{2}(ℝ × 𝕊^{n−1}), 0 ≤ j ≤ k}

⟶ H^{k}(B(ρ)) if k ≥ 0, and (33)

\begin{matrix}R^{t} : \{u ∈ C^{−∞}(ℝ × 𝕊^{n−1}); & u ∈ L_{loc}^{2}(ℝ × 𝕊^{n−1}) + D_{s}^{−k}L^{2}(ℝ × 𝕊^{n−1})\} \\ ⟶ H^{k}(B(ρ)) if k ≤ 0. \end{matrix} (34)

That is, one does not need tangential regularity to ensure the regularity of R_{n}^{t}f in a compact set.

Lemma 3. For any k ∈ ℤ satisfying \frac{n−1}{2} ≥ k ≥ −\frac{n+3}{2}, and any potential V ∈ Ḣ^{\frac{n+1}{2}} (B(ρ)), V_{LP} gives a bounded map

(35)  V_{LP} : H^{k}(ℝ × 𝕊^{n−1}) ⟶ H^{k+1}(ℝ × 𝕊^{n−1}).

Proof. Recall that V_{LP} = c_{n}^{2}Ds^{\frac{n−1}{2}} R · V · R^{t}Ds^{\frac{n−3}{2}} . From (33),

(36)  R^{t}Ds^{\frac{n−3}{2}} : H^{k}(ℝ × 𝕊^{n−1}) ⟶ H^{k+1}(B(ρ)).

Then, from Lemma 1, multiplication by V maps into the space Ḣ^{k+1}(B(ρ)) and

from Lemma 2, Ds^{\frac{n−1}{2}} R maps into Ḣ^{k+1}([−ρ, ρ] × 𝕊^{n−1}). □

3. Generalized backscattering transform

We shall apply these regularity estimates to show that a ‘modified backscattering transform,’ in which ‘excess’ information has been discarded, extends by continuity to Ḣ^{\frac{n+1}{2}} (B(ρ)).

Let π_{S,ρ} be the orthogonal projection, in H^{2}([−2ρ, 2ρ] × 𝕊^{n−1}), onto the closure

of the range of Ds^{\frac{n−3}{2}} R_{n} applied to (Id −S)^{∗}Ḣ^{\frac{n+1}{2}} (B(ρ)) = Ḣ^{\frac{n+1}{2}} ((Id −S)B(ρ)) using Lemma 2; let P_{S,ρ} be the range of π_{S,ρ}. For V ∈ C_{c}^{∞}(ℝ^{n}) we know that the scattering kernel κ_{V} , has support in {s ≥ −2ρ}. We will ‘cut off the tail’ where s > 2ρ and project the rest using π_{S,ρ}. Thus, consider the combined restriction, differentiation and projection map

(37) χ_{ρ} : C^{∞}(ℝ × 𝕊^{n−1}) ⟶^{D \frac{n−3}{2} ; s} C^{∞}([−2ρ, 2ρ] × 𝕊^{n−1})

⟶^{π_{S,ρ}} \overline{Ds^{\frac{n−3}{2}} R_{n}(Ḣ^{\frac{n+1}{2}} ((Id −S)B(ρ))} ⊂ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1}).

Now, for V ∈ C_{c}^{∞}(ℝ^{n}) we know that

(38)  singsupp κ_{V} ⊆ {s = 0, θ = ω}.

Thus the generalized backscattering kernel, κ_{V} (s, Sθ, θ) ∈ C^{∞}(ℝ × 𝕊^{n−1}). We can therefore apply (37) to define the modified (and generalized) backscattering transform

(39) β_{S} : Ċ^{∞}(B(ρ)) ∋ V ⟼ χ_{ρ}[κ_{V} (s, Sθ, θ)] ∈ P_{S,ρ} ⊂ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1}).

Theorem (Main Result). For any orthogonal transformation S, such that Id −S is invertible, the modified backscattering transform (39) extends, by continuity, to

(40)  β_{S} : Ḣ^{\frac{n+1}{2}} (B(ρ)) ⟶ P_{S,ρ} ⊂ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1})

which is entire analytic. More precisely, it can be written

(41)  β_{S}(V) = ∑_{j=1}^{∞} β_{S}^{j}(V_{,} … V)

where

(42)  β_{S}^{1} : Ḣ^{\frac{n+1}{2}} (B(ρ)) ⟶ P_{S,ρ} ⊂ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1})

is a linear isomorphism and for each j ≥ 2

(43)  β_{S}^{j} : [Ḣ^{\frac{n+1}{2}} (B(ρ))]^{j} ⟶ P_{S,ρ} ∩ Ḣ^{\frac{5}{2}} ([−2ρ, 2ρ] × 𝕊^{n−1})

is symmetric and satisfies, for each 0 ≤ ϵ ≤ \frac{1}{2},

(44)  ∥β_{S}^{j}(V, …, V)∥_{\frac{5}{2} −ϵ} ≤ \frac{C^{j+1}∥V ∥^{j}}{(j!)^{2ϵ}} .

As we shall describe below, this implies that β_{S} is almost everywhere a local isomorphism. It is not known, at least to the authors, whether β_{S} is a global isomorphism (for any admissible S, in particular S = − Id). Nor indeed is it known whether the differential of β_{S}, at V ∈ Ḣ^{\frac{n+1}{2}} (B(ρ)) is always invertible – although it is Fredholm. Nor is there a conjectural characterization of the singular points.

The Taylor expansion (41) for the modified backscattering transform is closely related to the Born approximation. This in turn is just the Neumann (or perhaps better to say Volterra) series for the solution of the (Radon-transformed) wave equation.

Formally at least, the solution to (21) can be expanded as a series

(45)  \begin{matrix}α = δ(s − t)δ_{θ}(ω) + ∑_{j=1}^{∞}(−1)^{j}α_{j}, α_{j} = [(D_{t} + D_{s})^{−1}V_{LP}]^{j}α_{0}, j ≥ 1 \\ α_{0} = δ(s − t)δ_{θ}(ω). \end{matrix}

Here (D_{t} + D_{s})^{−1} is the inverse of the free forcing problem

(46) (D_{t} + D_{s})u = f, f = 0 in s < −ρ, u = 0 in s < −ρ ⟹ u = (D_{t} + D_{s})^{−1}f.

We proceed to show that, for any V ∈ Ḣ^{\frac{n+1}{2}} (B(ρ)), the series (45) converges.

Proposition 1. For any V ∈ Ḣ^{\frac{n+1}{2}} (B(ρ)), T < ∞ and k ∈ ℤ, with −\frac{n+3}{2} ≤ k ≤ \frac{n+1}{2}, (D_{t} + D_{s})^{−1}V_{LP} is bounded as an operator on

(47)  Ḣ_{T,ρ}^{k} = {f ∈ Ḣ^{k}([−∞, T]_{t} × [−ρ, ρ]_{s} × 𝕊^{n−1}); f = 0 in t < −ρ}

and for some C = C(T)

(48)  ∥[(D_{t} + D_{s})^{−1}V_{LP}]^{j}∥_{H^{k}} ≤ \frac{C^{j+1}∥V ∥^{j}}{j!} ,

where ∥V ∥ is the norm in Ḣ^{\frac{n+1}{2}} (B(ρ)).

Proof. Since t is a parameter in the action of V_{LP} and (D_{t} + D_{s})^{−1} is bounded on any Sobolev space the boundedness is clear from Lemma 3. Only the Volterra-type estimate (48) needs to be shown. To carry out this estimation it is convenient to introduce D_{t} + D_{s} and D_{s} as coordinate vector fields, i.e. change coordinates to

(49)  t′ = t, s′ = s − t.

The operators are transformed as follows

(50)  D_{t} + D_{s} ⟼ D_{t′}, V_{LP} ⟼ V′_{LP}(t′, s′, D_{s′})

where V′_{LP} is still a non-local operator in s′, but now depending on t′ as a parameter, i.e.

(51)  V′_{LP}u(t′, s′) depends only on u(t′, ·).

The iterated operator is therefore

(52)  (D_{t′}^{−1}V′_{LP})^{j} .

Applying this |k| + 1 times to H^{k} gives a bounded map into the space

C^{0}([−ρ, T]; H^{k}(𝕊^{n−1} × ℝ_{s′})).

Then, integration in t′ and continuity of V′_{LP} shows that

(53)  ∥(D_{t′}^{−1}V′_{LP})^{j+|k|+1}u∥_{H^{k}(𝕊^{n−1}×ℝ_{s′})}(t′) ≤ \frac{C(t′ + ρ)^{j}}{j!} .

This gives (48). □

Of course from Lemma 3 we know that, if −\frac{n+3}{2} ≤ k ≤ \frac{n−1}{2},

(54)  (D_{t} + D_{s})^{−1}V_{LP} : Ḣ_{T,ρ}^{k} ⟶ Ḣ_{T,ρ}^{k+1}

Since

(55)  δ(t − s)δ_{θ}(ω) ∈ H_{loc}^{− \frac{n+1}{2}} (ℝ^{2} × 𝕊^{n−1} × 𝕊^{n−1})

it follows that

(56)  α_{j} ∈ H_{loc}^{− \frac{n+1}{2} +min(j,n+1)}(ℝ^{2} × 𝕊^{n−1} × 𝕊^{n−1}).

Consider the successive terms, α_{j}, in (45). Since V_{LP} always restricts supports to [−ρ, ρ] in s,

(57)  supp(α_{j}) ⊆ {t ≥ −ρ} ∩ {s ≥ −ρ} ∩ {t − s ≥ −2ρ} ∩ {t − s ≤ 2jρ}.

To get the expansion (41) we need to use (45) and then project each term with χ_{ρ}, after restricting to s = ρ, ω = Sθ (and shifting in t) to get the scattering kernel. Thus if

(58)  κ_{j}(s, ω, θ) = α_{j}(s − ρ, ρ, θ, ω)

then

(59)  β_{S}^{j}(V) = χ_{ρ}[κ_{j}(s, Sθ, θ)].

Since, as a function of t−s, s, ω and θ, α_{j} is independent of s in s > −ρ it follows from (56) that

(60)  κ_{j} ∈ H^{− \frac{n+1}{2} +min(j,n+1)}([−2ρ, T) × 𝕊^{n−1} × 𝕊^{n−1}) for any T.

Restricting to ω = Sθ, a submanifold of codimension n − 1 shows that

(61)  κ_{j}(s, Sθ, θ) ∈ H^{1}([−2ρ, T) × 𝕊^{n−1}) if j ≥ n + 1.

Moreover, to get (61) we only use the regularity property (54) for the first n + 1 factors in (52). Thus we conclude that the map

(62)  Ḣ^{\frac{n+1}{2}} (B(ρ)) ⟶ ∑_{j≥n+1} κ_{j}(s, Sθ, θ) ∈ H^{1}([−2ρ, T) × 𝕊^{n−1}) is entire

for each ρ. This is a good deal weaker than we need to prove the Theorem. Obviously we need to examine the first n + 1 terms in the Taylor series of β at V = 0 to show that this polynomial in V is defined and in any case we have to show that the whole map β_{S} takes values in H^{2} rather than H^{1}. Nevertheless we shall use (62) because it allows us to prove that β is entire, with values in the good space (essentially because of Pettit’s theorem).

4. Proof of the main result

First we examine

(63)  κ_{S}^{1}(s, Sθ, θ) = α_{1}(s − ρ, ρ, Sθ, θ).

This already has support in [−2ρ, 2ρ]. We wish to show that this, the linear, term is as claimed in (42). We proceed to compute κ_{1} explicitly. It is convenient to take the Fourier transform in s :

(64)  k̂_{1}(λ, ω, θ) = ∫_{−∞}^{∞} e^{−iλt}κ_{1}(t, ω, θ)dt = α̂_{1}(λ, ρ, ω, θ)e^{iλρ}.

¿From the definition of α_{1}, this gives

(65)  \begin{matrix}κ̂_{1}(λ, ω, θ) = e^{iλρ} ∫_{−∞}^{∞} ∫ e^{−iλ(ρ−s′)} [V_{LP}e^{−iλs}δ_{θ}(ω)] ds′. \\ = c_{n}^{2} ∫ e^{iλs}Ds^{\frac{n−1}{2}} ∫_{x·ω=s} V (x)λ^{\frac{n−3}{2}} e^{−iλx·θ}dxds. \end{matrix}

Integrating by parts we get

(66)  κ̂_{1}(λ, ω, θ) = c_{n}^{2}λ^{n−2} ∫ e^{iλx·(ω−θ)}V (x)dx.

Setting ω = Sθ we find

(67)  κ̂_{S}^{1}(λ, Sθ, θ) = c_{n}^{2}λ^{n−2}V̂ (λ(Id −S)θ).

Thus { \atop β_{S}^{1}(V)} is the (n-dimensional) Fourier transform of 2^{−n}V ((Id −S)^{−1}x) = Ṽ_{S}. Hence,

(68)  β_{S}^{1} = c_{n}Ds^{\frac{n−3}{2}} R_{n}Ṽ_{S}

shows that β_{S}^{1} maps into Ḣ^{2}([−2ρ, 2ρ]×𝕊^{n−1}). It is obviously an isomorphism onto

Ds^{\frac{n−3}{2}} R_{n}Ḣ^{\frac{n+1}{2}} ((Id −S)B(ρ)) (which is closed) as claimed.

Next we proceed to find a formula generalizing (66) to the higher derivatives at zero. From (57) we see that, for s bounded above, the support of each α_{j} is compact in t. After taking the Fourier transform in t, the iterative definition (45) becomes

(69)  α̂_{j}(λ, s, ω, θ) = (D_{s} + λ)^{−1}R_{n}[V · Q_{λ}]^{j−1}V R^{t}D_{s}^{(n−3)/2}e^{−isλ}δ_{θ}(ω),

where

(70)  Q_{λ} = R_{n}^{t}D_{s}^{−1}(D_{s} + λ)^{−1}R_{n}.

Here D_{s}^{−1}, and (D_{s}+λ)^{−1} mean integration from s = −∞, i.e. the inverse preserving vanishing to the left.

Lemma 4. Acting from C_{c}^{∞}(ℝ^{n}) to C^{∞}(ℝ^{n}), Q_{λ} = (Δ − λ^{2})^{−1} is the analytic extension of the ‘free resolvent’ defined as a bounded operator on L^{2} for ℑλ < 0.

Proof. This formula can be deduced from the modified Radon transform of Lax and Phillips. We know that this intertwines the wave group U_{0}(t) with the translation group, so conjugates the infinitesimal generator of one to that of the other

(71)  c_{n}(Ds^{\frac{n−1}{2}} R, Ds^{\frac{n+1}{2}} R) (\begin{matrix}0 & −1 \\ Δ & 0 \end{matrix}) = D_{s}(Ds^{\frac{n−1}{2}} R, Ds^{\frac{n+1}{2}} R).

For ℑλ < 0, so in the resolvent set, it follows that

(72)  c_{n}^{2}R^{t}Ds^{\frac{n−3}{2}} (D_{s} + λ)^{−1}Ds^{\frac{n−1}{2}} = (Δ − λ^{2})^{−1}.

This proves the lemma. □

Inserting the integral expression for (D_{s} + λ)^{−1} into (69) gives

(73) α̂_{j}(λ, s, ω, θ) = c_{n}^{2} ∫_{−∞}^{s} e^{−iλ(s−s′)}Ds′^{\frac{n−1}{2}} ∫_{x·ω=s′} V · Q_{λ} · V ⋯

Q_{λ} · [V (•)(−λ)^{\frac{n−3}{2}} e^{−iλ•·θ}]dH_{x}ds′.

¿From (58), by setting s = ρ and integrating by parts we find

(74) κ̂_{j}(λ, ω, θ) = c_{n}^{2}(−1)^{\frac{n−3}{2}} λ^{n−2} ∫_{ℝ^{n}} e^{iλω·x}V (x)[Q_{λ} ⋯ Q_{λ} · V (•)e^{−iλθ·•}](x)dx.

Restricting to backscattering, ω = Sθ, this gives κ̂_{S}^{j} in a form similar to (67). Since κ_{j} has support in [−2ρ, 2jρ] its regularity can be deduced from its Fourier-Laplace transform with ℑλ = −1. Thus we need to examine the growth in λ of

(75) κ̂_{j}(λ, Sθ, θ) =

c_{n}^{2}λ^{n−2} ∫_{ℝ^{jn}} e^{iλθ·(S^{t}x^{(1)}−x^{(j)})}V (x^{(1)})Q_{λ}(x^{(1)} − x^{(2)})V (x^{(2)}) …

… Q_{λ}(x^{(j−1)} − x^{(j)})V (x^{(j)})(x)dx^{(1)} … dx^{(j)}

where there are j − 1 factors of the free resolvent, Q_{λ}, and j factors of V. As a convolution operator Q_{λ} has kernel

(76)  Q_{λ}(y) = (2π)^{−n} ∫ e^{iy·η}(|η|^{2} − λ^{2})^{−1}dη.

Inserting this into (75) gives

(77) κ̂_{j}(λ, Sθ, θ) =

c_{n}^{2} ∫ V (x^{(1)})V (x^{(2)}) … V (x^{(j)}) ∏_{ℓ=1}^{j−1}(|η^{(ℓ)}|^{2} − λ^{2})^{−1}

× exp[i(S^{t}x^{(1)} − x^{(j)}) · ξ + i(x^{(1)} − x^{(2)}) · η^{(1)} + ⋯ + i(x^{(j−1)} − x^{(j)}) · η^{(j−1)}]

dx^{(1)} … dx^{(j−1)}dη^{(1)} … dη^{(j−1)}

where ξ = λθ.

Carrying out the x-integrals in (77) gives (78)

κ̂_{j}(λ, Sθ, θ) ; \begin{aligned}= c_{n}^{2}λ^{n−2} & ∫ V̂ (−S^{t}ξ − η^{(1)})V̂ (η^{(1)} − η^{(2)}) … V̂ (η^{(j−2)} − η^{(j−1)})V̂ (η^{(j−1)} + ξ) \\ & ∏_{ℓ=1}^{j−1}(|η^{(ℓ)}|^{2} − λ^{2})^{−1}dη^{(1)} … dη^{(j−1)}. \end{aligned}

Apart from the factors arising from the resolvent this is an iterated convolution. Since ℑλ = −1, the resolvent factors are non-singular. Using the obvious estimates

(79)  |(|η|^{2} − λ^{2})^{−1}| ≤ c(1 + |η| + |λ|)^{−1}.

and

(80)  (1 + |η′| + |λ|)^{−1}(1 + |η| + |λ|)^{−1} ≤ (1 + |η − η′|)^{−1}

the right side of (78) can be estimated to give

(81)  |κ̂_{j}(λ, Sθ, θ)| ≤ C^{j+1}|λ|^{n−2}× ; \begin{matrix}∫ Φ̂ & (−S^{t}ξ − η^{(1)})Φ̂(η^{(1)} − η^{(2)}) \\ … Φ̂(η^{(j−2)} − η^{(j−1)})Φ̂(η^{(j−1)} + ξ)dη^{(1)} … dη^{(j−1)}, \end{matrix}

where

(82)  Φ̂(η) = |V̂ (η)|(1 + |η|)^{− \frac{1}{2}} .

Thus

(83)  ∥Φ∥_{H^{(n+2)/2}} ≤ ∥V ∥_{H^{(n+1)/2}}.

First translating the variables of integration to η^{(ℓ)} +ξ we find that the right side of (81) is the Fourier transform of a product of functions, so using Lemma 1 repeatedly (and taking into account the factor of λ^{n−2} and the invertibility of S^{t} − Id)

(84)  ∥κ_{j}(s, Sθ, θ)∥_{H^{\frac{5}{2}} ([−2ρ,2ρ]×𝕊^{n−1}} ≤ C^{1+j}∥V ∥_{H^{(n+1)/2}}.

This gives the desired continuity (43) and estimates (44) for ϵ = 0. Moreover the estimates (66) give (44) for ϵ = \frac{1}{2} and large (hence all) j. The estimates for all ϵ ∈ [0, \frac{1}{2}] then follow by interpolation between Sobolev spaces, i.e.

(85)  ∥u∥_{\frac{5}{2} −ϵ} ≤ C∥u∥_{2}^{2ϵ}∥u∥1−2ϵ_{\frac{5}{2}} ∀ ϵ ∈ [0, \frac{1}{2}].

This completes the proof of Theorem 3.

It may be that the estimates centered on (79) can be improved to give the exponential type estimates (44) directly and with values in H^{\frac{5}{2}} . If the original regularity (n + 1)/2 for V is increased by p then the regularity of the derivatives β_{S}^{j} in (44) can also be increased by p.

Note that the map β_{S} in (59) is defined by projection onto the range of the linearization of V ⟼ κ_{V} (s, Sθ, θ) at V = 0. The linearization has been shown to be an injective Fredholm map, i.e. is an isomorphism onto its (closed) range, so its generalized inverse is a bounded map

(86) L_{S} : Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1}) ⟶ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1}) ⟶ Ḣ^{\frac{n+1}{2}} (B(ρ)).

The map in (1) is then

(87)  L_{S}β_{S}(V) = L_{S}(κ_{V} (s, Sθ, θ)) on Ḣ^{\frac{n+1}{2}} (B(ρ))

which is therefore an entire map with linearization the identity at 0 and derivative at all other points a compact perturbation of the identity.

5. Fredholm property

Proposition 2. There is a closed subset G(ρ) ⊂ Ḣ^{\frac{n+1}{2}} (B(ρ)) which is of codimension at least two (i.e. locally orthogonal projection from G(ρ) onto some subspace of codimension two is at most p-to-1 for some fixed p ∈ ℕ) such that for each V′ ∈ [Ḣ^{\frac{n+1}{2}} (B(ρ)) \ G(ρ)] there exists ϵ > 0 such that the map

(88) β_{S} : \{V ∈ Ḣ^{\frac{n+1}{2}} (B(ρ)); ∥V − V′∥ < ϵ\} ⟶ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1})

is an isomorphism onto its image.

Proof. The set G(ρ) is defined to consist of those V ∈ Ḣ^{\frac{n+1}{2}} (B(ρ)) such that the derivative of β_{S} with respect to V is not an isomorphism. Certainly (88) holds for points in the complement of G(ρ) by the implicit function theorem, applied in the Sobolev space. Thus we need to show that G(ρ) so defined has codimension at least 2, since the density of the complement certainly follows from this. The derivative of β_{S} with respect to V is a linear map

(89)  β_{1} + γ(V) : Ḣ^{\frac{n+1}{2}} (B(ρ)) ⟶ Ḣ^{2}([−2ρ, 2ρ] × 𝕊^{n−1})

where β_{1} is an isomorphism and γ(V) depends analytically on V and maps continuously into Ḣ^{\frac{5}{2} −ϵ}([−2ρ, 2ρ] × 𝕊^{n−1}). If we consider simply the complex multiples of V, i.e. just look at γ(zV), we have analyticity in z. The invertibility of this operator reduces to a finite dimensional problem. Since the map is known to be invertible at z = 0, invertibility can only fail at isolated values of z. This proves the result. □

Corollary. For each ρ > 0 there is a dense subset of Ċ^{∞}(B(ρ)) near each point of which the backscattering transform (88) is injective from Ċ^{∞}(B(ρ)).

References

[Ad] R. Adams, Sobolev spaces, Vol. 65., Pure and Applied Mathematics, Academic Press, 1975.

[BLM] A. Bayaliss, Y. Li and C. S. Morawetz, Scattering by a potential using hyperbolic methods, Math. Comp. 52(1989), 321–338.

[BM] I. Beltita and A. Melin, Local smoothing for the backscattering transform, arXiv:0712:3865v1.

[ER1] G. Eskin and J. Ralston, The inverse backscattering problem in three dimensions, Comm. Math. Phys., 124(1989), 169–215.

[ER2] G. Eskin and J. Ralston, The inverse backscattering problem in two dimensions, Comm. Math. Phys., 138(1991), 451–486.

[ER3] G. Eskin and J. Ralston, Inverse Backscattering, J. Anal. Math., 58(1992), 177–190.

[GU] A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularities of scattering data, Comm. Math. Phys., 157 (1993), 549–572.

[Hel] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math., 113(1965), 153–180.

[J] M. Joshi, Recovering the total singularity of a conormal potential from backscattering data, Ann. Inst. Fourier,48(1998), 1513-1532.

[LP] P. Lax and R. Phillips, Scattering Theory, Revised edition, Academic Press, 1989.

[Me1] A. Melin, Back-scattering and nonlinear Radon transformation, Séminaire Équations aux Dérivées Partielles, École Polytechnique, Palaiseau (1998-1999).

[Me2] A. Melin, Some transforms in potential scattering, Contemp. Math., 348(2003), 103-134.

[Mel] R. B. Melrose, Geometric Scattering Theory, Cambridge University Press (1995).

[MU] R. Melrose and G. Uhlmann, Introduction to microlocal analysis, lecture notes available.

[Mo] C. S. Morawetz, A formulation for higher dimensional inverse problems for the wave equation, Comp. & Math. Appl., 7(1981), 319–331.

[OPS] P. Ola, L. Päivärinta and V. Serov, Recovering singularities from backscattering in two dimensions, Comm. P.D.E., 26(2001), 697-715.

[P] V. Petkov, Scattering Theory for Hyperbolic Operators, North-Holland, (1989).

[R] A. Ruiz, Recovery of the singularities of a potential from fixed angle scattering data, Comm. P.D.E., 26(2001), 1721-1738.

[RV] A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. P.D.E., 30(2005), 67-96.

[S] P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. P.D.E., 17(1992), 55–68.

[StU] P. Stefanov and G. Uhlmann, Inverse backscattering for the acoustic equation, SIAM J. Math. Anal., 28 (1997), 1191-1204.

[SuU] Z. Sun and G. Uhlmann, Recovery of singularities for formally determined inverse problems, Comm. Math. Phys., 153(1993), 431-445.

[U] G. Uhlmann, A time dependent approach to the inverse backscattering problem, Inverse Problems, 17(2001), 703-716.

[W] J-N Wang, Inverse backscattering in even dimensions, Math. Z., 239(2002), 365-379.

Department of Mathematics, Massachusetts Institute of Technology

   E-mail address: rbm@math.mit.edu

Department of Mathematics, University of Washington

   E-mail address: gunther@math.washington.edu