THE CALDERÓN PROBLEM FOR CONORMAL POTENTIALS,

I: GLOBAL UNIQUENESS AND RECONSTRUCTION

Allan Greenleaf, Matti Lassas and Gunther Uhlmann

Abstract. We consider inverse problems for Schrödinger operator in the case of potential having co-normal singularities in dimensions n ≥ 3. For instance, in the case where singularity is on hypersurface H ⊂ ℝ^{n}, we allow singularities of type dist(x, H)^{−1+α}, α > 0. Moreover, by considering Schrödinger equation in generalized sense (as solution of minimization problem), we give certain counter examples to uniqueness of inverse problem.

1. Introduction

The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in ℝ^{n}. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function δ_{H} on the hypersurface H; in general, these potentials correspond to conductivities which are in C^{1+ϵ} and thus fail to be covered by previously known results.

Let Ω ⊂ ℝ^{n} be a bounded Lipschitz domain, H ⊂ Ω a smooth submanifold of codimension k, and q ∈ I^{μ}(H) a real conormal distribution of order μ with μ < 1 − k. Thus, if H = {x : F_{j}(x) = 0, 1 ≤ j ≤ k} is a local representation of H by means of defining functions with {∇F_{j} : 1 ≤ j ≤ k} linearly independent on H, then locally q(x) has the Fourier integral representation

(1.1)  q(x) = ∫_{ℝ^{k}} e^{i ∑_{j} F_{j}(x)·θ_{j}}a(x, θ)dθ, a ∈ S_{1,0}^{μ},

where S_{1,0}^{μ} denotes the standard class of symbols of order μ and type (1, 0) on ℝ^{n}×(ℝ^{k}\0). A general element q ∈ I^{μ}(H) is a locally finite sum of such expressions. We assume throughout that supp(q) is compact in Ω. If −k < μ < 0, then q satisfies |q(x)| ≤ C · dist (x, H)^{−k−μ}, so that q ∈ L^{\frac{k}{k+μ} −ϵ}(Ω), ∀ϵ > 0, and no better in general; in particular, a general element of I^{μ}(H) is unbounded. For comparison, surface measure δ_{H} ∈ I^{0}(H) and, in the hypersurface case, a Heaviside discontinuity across H belongs to I^{−1}(H).

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This work was supported in part by National Science Foundation grants DMS-9877101, DMS-0070488 and Finnish Academy project 172434. The third author was also supported by a John Simon Guggenheim Fellowship. The second and third author are grateful for the hospitality of the Mathematical Sciences Research Institute in Berkeley, CA.

Rather than working with the Dirichlet-to-Neumann map, Λ_{q}, we state our main results in terms of the Cauchy data, CD_{q}, of sufficiently regular solutions of the Schrödinger equation

(1.2)  (Δ + q(x))u(x) = 0 on Ω.

This is more flexible, since CD_{q} can be defined for potentials q for which Λ_{q} is either not defined (for example, if λ = 0 is a Dirichlet eigenvalue) or is not known to be defined (due to the low regularity of q(x)); it is perhaps more natural as well.

It will be convenient to write μ = ν − k. Assume that ν_{0}(k) < ν < 1, where ν_{0}(k) =^{def} max(\frac{2}{3}, 1 − \frac{k}{4} ). Fix p and r satisfying

(1.3)  2 ≤ r < \frac{k}{2(1 − ν)} =^{def} r_{0}(k, ν) < p_{0}(k, ν) =^{def} \frac{2k}{k − ν} < p < ∞.

(If ν ≤ ν_{0}(k), just pick p and r for some ν′ > ν_{0}(k).) Fixing a smooth function ψ ∈ C^{∞}(ℝ^{n}), ψ ≡ 1 near ∂Ω and int{ψ = 0} ∩ Ω ≠ ϕ, define the norm

(1.4)  ||f||_{X^{p,r}} = ||f||_{L^{p}(Ω)} + ||Δf||_{L^{p′} (Ω)} + ||ψf||_{W^{2,r}(Ω)},

where p′ is the dual exponent to p and W^{2,r} is the standard Sobolev space of f ∈ D′(Ω) having two derivatives in L^{r}(Ω). Set

(1.5)  X^{p,r}(Ω) = {f ∈ D′(Ω) : ||f||_{X^{p,r}} < ∞}

and note that the Schrödinger operator Δ+q maps X^{p,r}(Ω) → L^{p′}(Ω) continuously. We denote throughout this paper by n the unit outer normal to Ω.

Definition. For a potential q ∈ I^{μ}(H) with H ∩supp(q) ⊂ int{ψ = 0}, the Cauchy data of the Schrödinger operator Δ + q relative to X^{p,r}(Ω) is

(1.6)  CD_{q} = CD_{q}^{p,r} = \{(u|_{∂Ω}, \frac{∂u}{∂n}|_{∂Ω}) : u ∈ X^{p,r}(Ω), (Δ + q)u = 0 on Ω\}.

By Sobolev embedding, CD_{q} is a subspace of W^{2− \frac{1}{r} ,r}(∂Ω) × W^{1− \frac{1}{r} ,r}(∂Ω). Observe that if the Dirichlet-to-Neumann map Λ_{q} is defined (on W^{2− \frac{1}{r} ,r}(∂Ω), say), then CD_{q} is simply its graph. We will construct certain nontrivial exponentially growing solutions u ∈ X^{p,r}(Ω), so that, for the potentials considered, CD_{q} is in fact nontrivial. We can now state our first result.

Theorem 1. Suppose that for j = 1, 2, H_{j} ⊂⊂ Ω are submanifolds of codimension k_{j}. Suppose further that q_{j} ∈ I^{μ_{j}}(H_{j}) are real potentials with ν_{0}(k_{j}) − k_{j} < μ_{j} < 1 − k_{j} and supp(q_{j}) ⊂⊂ Ω. Let p, r satisfy 2 ≤ r < min(r_{0}(k_{1}, ν_{1}), r_{0}(k_{2}, ν_{2})) and max(p_{0}(k_{1}, ν_{1}), p_{0}(k_{2}, ν_{2})) < p < ∞ and suppose that ψ ≡ 0 on a neighborhood of H_{1} ∪ H_{2}. Then CD_{q_{1}} = CD_{q_{2}} relative to X^{p,r}(Ω) implies that q_{1} = q_{2} on Ω.

We also show that under the same assumptions as in Theorem 1 for the potential we have a reconstruction procedure, that is we can reconstruct q from CD_{q} (see Theorem 2 in section 3 for more details).

Global uniqueness was established in [32] for n ≥ 3 (for smooth potentials) and [26] for n = 2; for n ≥ 3 this was extended to q ∈ L^{∞} in [27]. The regularity was further lowered to q ∈ L^{\frac{n}{2}} in unpublished work of R. Lavine and A. Nachman and to potentials of small norm in the Fefferman-Phong class in [4]. Note that for −\frac{n−2}{n}k ≤ μ < 0, k < \frac{n}{2}, a general element of I^{μ}(H) fails to be in L^{\frac{n}{2}} (Ω).

The isotropic conductivity problem, where one considers the Dirichlet-to-Neumann map for L_{γ} = ∇(γ · ∇), can be reduced to the Schrödinger problem via the substi-

tution q = −\frac{Δ(γ^{\frac{1}{2}} )}{γ^{\frac{1}{2}}}, and thus the analogue of the Theorem holds for conductivities

γ_{j} ∈ I^{−k−1−ϵ}(H_{j}) ↪ C^{1+ϵ}(\overline{Ω}), ∀0 < ϵ < 1. Currently, the best general global uniqueness result known for n ≥ 3 is for γ ∈ C^{\frac{3}{2}} , proved in [28], building on [2] and using the general argument of [32], while the best known result for n = 2 is γ ∈ W^{1,p}(Ω), p > 2, proved in [3] using the \overline{∂} technique of [1,25,26]. Global uniqueness for piecewise-analytic conductivities was proven [20], and special types of jump discontinuities were treated in [17].

Here, we will follow the general argument of [32], although employing a different integral identity so as to avoid difficulties when applying Green’s Theorem. It is this identity that makes X^{p,r} a convenient space for the problem; indeed, both sides of ∫_{Ω} Δu · v − u · Δvdx = ∫_{∂Ω} ∂_{n}u · v − u · ∂_{n}vdσ are continuous with respect to || · ||_{X^{p,r}} and thus Green’s Theorem holds for u, v ∈ X^{p,r}(Ω).

We now start the proof of Thm. 1, so as to motivate the technicalities that follow.

Given a submanifold H of codimension k and a potential q ∈ I^{μ}(H) with μ < 1− k, we will construct exponentially growing solutions of (1.2) belonging to X^{p,r}(Ω), of the form v(x) = e^{ρ·x}(1+ψ(x, ρ)), with ρ ∈ ℂ^{n} satisfying ρ·ρ = 0. Let v_{1} ∈ X^{p,r}(Ω) be a solution of (Δ + q_{1})v_{1} = 0. By the hypothesis of Thm. 1, there is a solution v_{2} ∈ X^{p,r}(Ω) of (Δ + q_{2})v_{2} = 0 with

(1.7)  v_{2}|_{∂Ω} = v_{1}|_{∂Ω} and \frac{∂v_{2}}{∂n} |_{∂Ω} = \frac{∂v_{1}}{∂n} |_{∂Ω}.

Let w_{2} ∈ X^{p,r}(Ω) be any other solution to (Δ + q_{2})w_{2} = 0. Then,

(Δ + q_{2})(v_{1} − v_{2}) = (Δ + q_{2})v_{1} = (Δ + q_{1} + (q_{2} − q_{1}))v_{1} = (q_{2} − q_{1})v_{1},

so that

(1.8)

∫_{Ω}(q_{2} − q_{1})v_{1}w_{2}dx = ∫_{Ω}(Δ + q_{2})(v_{1} − v_{2}) · w_{2}dx
= ∫_{Ω}(Δ + q_{2})(v_{1} − v_{2}) · w_{2} − (v_{1} − v_{2}) · (Δ + q_{2})w_{2}dx
= ∫_{Ω} Δ(v_{1} − v_{2}) · w_{2} − (v_{1} − v_{2}) · Δw_{2}dx
= ∫_{∂Ω} \frac{∂}{∂n}(v_{1} − v_{2}) · w_{2} − (v_{1} − v_{2}) · \frac{∂}{∂n}w_{2}dσ
=0,

where the application of Green’s Theorem is valid since v_{1} − v_{2} and w_{2} ∈ X^{p,r} and the last equality holds by (1.7). If we carry this out for the solutions v_{1} and w_{2} constructed below for complex frequencies ρ_{1} and ρ_{2} satisfying ρ_{1} + ρ_{2} = −iξ, with ξ ∈ ℝ^{n}\0, then we have, as in [32],

(1.9)  \begin{matrix}0 = & ∫_{Ω}(q_{1} − q_{2})e^{(ρ_{1}+ρ_{2})·x}(1 + ψ_{1}(x, ρ_{1}))(1 + ψ_{2}(x, ρ_{2}))dx \\ = & { \atop (q_{1} − q_{2})}(ξ) + ∫_{Ω} e^{−iξ·x}(q_{1} − q_{2})(ψ_{1} + ψ_{2} + ψ_{1}ψ_{2})dx \end{matrix} .

If one can do this for pairs (ρ_{1}, ρ_{2}) with |ρ_{j}| → ∞ and show that the last integral → 0 as |ρ| → ∞, then q̂_{1}(ξ) = q̂_{2}(ξ); doing this for all ξ ∈ ℝ^{n} will finish the proof of Thm. 1.

As is well known, v(x) = e^{ρ·x}(1 + ψ(x)) is a solution of the Schrödinger equation iff ψ is a solution of

(1.10)  (Δ_{ρ} + q)ψ = −q(x) where Δ_{ρ} = Δ + 2ρ · ∇.

We will show in Prop. 2.6 that (1.10) is uniquely solvable, with some decay in |ρ|, in a Banach space of finite-regularity conormal distributions associated with H, yielding exponentially growing solutions v_{j} ∈ X^{p,r} to (1.2) which allow the argument above to be carried out. In §3, this result is extended to a hybrid global space, and this is applied to obtain reconstruction of the potential from the Cauchy data, following the general argument of [25]. Finally, in §4 we show that uniqueness can fail in a weak formulation of the problem for potentials with very strong singularities on a hypersurface, with blow-up rates corresponding to those of distributions conormal of order greater than 1 for H.

We would like to thank Steve McDowall for valuable discussions.

2. Uniqueness for conormal potentials

As described in the Introduction, to prove Thm.1, it suffices to construct exponentially growing solutions to (1.2) of the form v(x) = e^{ρ·x}(1 + ψ(x, ρ)) for ρ · ρ = 0, |ρ| → ∞ so that the second integral in (1.9) tends to 0 as |ρ| → ∞. To do this for potentials q ∈ I^{μ}(H), the standard space of (infinite-regularity) conormal distributions of order μ associated with the codimension k submanifold H, we will also need to formulate Banach spaces of finite-regularity conormal distributions in ℝ^{n} associated with H. Rather than working in unnecessary generality, we will restrict ourselves to the spaces needed here; unlike [23],[22], where several other types of finite-regularity conormal spaces are defined using iterated regularity with respect to Lie algebras of tangent vector fields, we impose the finite-regularity assumption directly on the symbols in the oscillatory representations of the distributions.

Fix a compact set K with Ω ⊂⊂ K; all of the spaces will involve distributions supported in K. Also fix an order m ∈ ℝ, a sequence δ⃗ = (δ_{1}, δ_{2}, ...) of numbers 0 ≤ δ_{j} ≤ 1, and N_{x}, N_{θ} ∈ ℕ.

Definition 2.1. (i)

S_{N_{x},N_{θ}}^{m,δ⃗} = {a(x, θ) : |∂_{x}^{β}∂_{θ}^{α}a(x, θ)| ≤ C(1 + |θ|)^{m−∑_{j=1}^{|α|} δ_{j}}, ∀|β| ≤ N_{x}, |α| ≤ N_{θ}} and

||a||_{S_{Nx,N_{θ}}^{m,δ⃗}} = max_{0≤α≤N_{θ}^{0≤|β|≤Nx}} sup_{x∈K,θ∈ℝ^{k}}(1 + |θ|)^{−m+∑_{j=1}^{|α|} δ_{j}}|∂_{x}^{β}∂_{θ}^{α}a(x, θ)|.

(ii) If H = {x : F_{j}(x) = 0, 1 ≤ j ≤ k} locally, with {∇F_{j}} linearly independent on H, then

I_{N_{x},N_{θ}}^{m,δ⃗}(H) = {u(x) = ∫_{ℝ^{k}} e^{i ∑_{j=1}^{k} F_{j}(x)·θ_{j}}a(x, θ)dθ : a ∈ S_{N_{x},N_{θ}}^{m,δ⃗}}

locally and in general consists of all locally finite sums of such expressions.

Remarks. (1) S_{N_{x},N_{θ}}^{m,δ⃗} is a Banach space with respect to the norm defined in (i),

and I_{N_{x},N_{θ}}^{m,δ⃗}(H) inherits this structure.

(2) ∂_{θ} : S_{N_{x},N_{θ}}^{m,(δ_{1},δ_{2},...)} → S_{N_{x},N_{θ}−1}^{m−δ_{1},(δ_{2},δ_{3},...)} and ∂_{x} : S_{N_{x},N_{θ}}^{m,(δ_{1},δ_{2},...)} → S_{N_{x}−1,N_{θ}}^{m,(δ_{1},δ_{2},...)} continuously.

(3) I_{N_{x},N_{θ}}^{m,δ⃗}(H) is well-defined, since changing the defining function of H locally corresponds to a change of variable in x, which leaves the symbol class invariant.

(4) The usual (infinite-regularity) conormal space has a continuous inclusion with respect to its Fréchet structure: I^{m}(H) ↪ I_{N_{x},N_{θ}}^{m,δ⃗}(H) for any δ⃗, N_{x}, N_{θ}.

Proposition 2.2. Let H ⊂ Ω ⊂ ℝ^{n}, with codim(H) = k.

(i) For −k ≤ m < −\frac{k}{2} , N_{x} ≥ 1, N_{θ} ≥ 1, δ_{1} > 0 and m − ∑_{j=1}^{N_{θ}} δ_{j} < −k,

(2.1)  I_{N_{x},N_{θ}}^{m,δ⃗}(H) ↪ L^{p}(Ω) continuously for all 1 ≤ p < \frac{k}{m + k} .

(ii) If N_{x} ≥ 2, N_{θ} ≥ 2, δ_{1} > 1 − \frac{k}{2} and ∑_{j=1}^{N_{x}} δ_{j} > 1, then for each smooth function F vanishing on H,

(2.2)  v ∈ I_{N_{x},N_{θ}}^{−k,δ⃗}(H) ⟹ F · v ∈ W^{1,p}(Ω) for 1 ≤ p < \frac{k}{1 − δ_{1}}

and thus, for G ⊂⊂ \overline{Ω}\H, I_{N_{x},N_{θ}}^{−k,δ⃗}(H) ↪ W^{1,p}(G) continuously. (iii) Suppose ν_{0}(k) < ν < 1, N_{x} ≥ 3 and N_{θ} ≥ 6. Set

δ⃗_{0} = (ν, 1 − ν, 2ν − 1, 1 − ν, 1 − ν, 3ν − 2, …).

Then, for any F vanishing on H,

(2.3)  v ∈ I_{N_{x},N_{θ}}^{−k,δ⃗_{0}}(H) ⟹ F^{3} · v ∈ W^{2,r}(Ω) for 1 ≤ r < \frac{k}{2(1 − ν)}

and thus, for G ⊂⊂ \overline{Ω}\H, I_{N_{x},N_{θ}}^{−k,δ⃗_{0}}(H) ↪ W^{2,r}(G) continuously.

Proof. (i) Since L^{p} and I_{N_{x},N_{θ}}^{m,δ⃗}(H) are diffeomorphism-invariant, it suffices to assume that, with respect to coordinates x = (x′, x′′) ∈ ℝ^{n−k} × ℝ^{k},

H = {x′′ = 0} and u(x) = ∫_{ℝ^{k}} e^{ix′′·θ}a(x, θ)dθ, a ∈ S_{1,1}^{m,δ⃗},

with S_{1,1}^{m,δ⃗} as in Def.2.1(i). We then have

u(x′, x′′) = ∫ e^{ix′′·θ}[a(x′, 0, θ)+ ∑_{0<|α′′|<N_{x}} \frac{1}{(α′′)!}∂_{x′′}^{α′′}a(x′, 0, θ)(x′′)^{α′′}+R_{N_{x}}(x′, x′′, θ)]dθ,

where

R_{N_{x}}(x, θ) = ∑_{|α′′|=N_{x}} b_{α′′}(x, θ)(x′′)^{α′′} with |b_{α′′}(x, θ)| ≤ C(1 + |θ|)^{m}, ∀α′′.

Since a(x′, 0, ·) ∈ L^{q}(ℝ_{θ}^{k}), ∀q > −\frac{k}{m} and −\frac{k}{m} < 2, the Hausdorff-Young inequality implies that ∫ e^{ix′′·θ}a(x′, 0, θ)dθ ∈ L^{q′}(ℝ_{x′′}^{k}), ∀2 ≤ q′ < \frac{k}{m+k} , uniformly in x′, and thus belongs to L^{p}(ℝ^{n}) for all 2 ≤ p < \frac{k}{m+k}; since it is compactly supported, the range is in fact 1 ≤ p < \frac{k}{m+k}. For the second term, note that for each function

a_{α′′}(x′, θ) =^{def} \frac{1}{(α′′)!}∂^{α′′}a(x′, 0, θ),

∫ e^{ix′′·θ}a_{α′′}(x′, θ)(x′′)^{α′′}dθ = ∫ (\frac{1}{i}∂_{θ})^{α′′}(e^{ix′′·θ})a_{α′′}(x, θ)dθ =

= ∫ e^{ix′′·θ}(\frac{1}{i}∂_{θ})^{α′′}(a_{α′′}(x, θ))dθ,

whose amplitude is ≤ C(1 + |θ|)^{m−∑_{j=1}^{|α′′|} δ_{j}}, and hence is treated by the same argument as for a_{0} = a(x′, 0, θ). In the final term, we integrate by parts:

∫ e^{ix′′·θ}b_{α′′}(x, θ)(x′′)^{α′′}dθ = ∫ e^{ix′′·θ}(i∂_{θ})^{α′′}b_{α′′}(x, θ)dθ,

and since |∂_{θ}^{α′′}b(x, θ)| ≤ C(1 + |θ|)^{m−∑_{j=1}^{N_{θ}} δ_{j}} ∈ L^{1}(ℝ_{θ}), uniformly in x′, this yields a bounded function of x.

(ii) If v ∈ I_{N_{x},N_{θ}}^{−k,δ⃗}(H), then v ∈ L^{p}, ∀p < ∞, by part (i). From

v(x) = ∫ e^{i ∑_{j=n−k+1}^{n} x_{j}θ_{j}}a(x, θ)dθ, a ∈ S_{N_{x},N_{θ}}^{−k,δ⃗},

one finds

\begin{matrix}∂_{x_{l}}v(x) = & ∫ e^{i ∑_{j=n−k+1}^{n} x_{j}θ_{j}}(iθ_{l}a + ∂_{x_{l}}a)dθ \\ ∈ & I_{N_{x},N_{θ}}^{1−k,δ⃗}(H) + I_{N_{x}−1,N_{θ}}^{−k,δ⃗}(H). \end{matrix}

The second term is covered by part (i) and hence is in L^{p}, ∀p < ∞. If the first term is multiplied by some x_{j_{0}}, n − k + 1 ≤ j_{0} ≤ n and integrated by parts, it becomes an element of I_{N_{x},N_{θ}−1}^{1−k−δ_{1},(δ_{2},δ_{3},...)}(H), which by (2.1) is in L^{p}, 1 ≤ p < \frac{k}{1−δ_{1}} . Since any F vanishing on H can be represented as a linear combination of x_{j_{0}}’s with smooth coefficients, F (x)∇v ∈ L^{p} and so v ∈ W^{1,p}(G) for any set G on which |F | is bounded below.

(iii) By (i) and (ii) above, both v and F · ∇v ∈ L^{r}. Now, arguing as in (ii),

\begin{matrix}∂_{x_{l}x_{l′}}^{2} v = & ∫ e^{i ∑_{j=n−k+1}^{n} x_{j}θ_{j}}(−θ_{l}θ_{l′}a + i(θ_{l}a_{x_{l′}} + θ_{x_{l′}} a_{x_{l}}) + a_{x_{l}x_{l′}} )dθ \\ ∈ & I_{N_{x},N_{θ}}^{2−k,δ⃗_{0}}(H) + I_{N_{x}−1,N_{θ}}^{1−k,δ⃗_{0}}(H) + I_{N_{x}−2,N_{θ}}^{−k,δ⃗_{0}}(H) \end{matrix}

By (i), the last term is in L^{r}, ∀r < ∞, if N_{x} ≥ 3, N_{θ} ≥ 1. Integrating by parts and using (ii), x_{j_{0}} times the second term is in L^{r}, 1 ≤ r < \frac{1}{1−δ_{1}} = \frac{1}{1−ν} if N_{x} ≥ 2, N_{θ} ≥ 2. As for the first term, x_{j_{0}}x_{j_{1}}x_{j_{2}} times it is seen, after integrating by parts three times, to be in I_{N_{x},N_{θ}−3}^{2−k−(δ_{1}+δ_{2}+δ_{3}),(δ_{4},...)}(H) = I_{N_{x},N_{θ}−3}^{2−k−2ν,(1−ν,1−ν,3ν−2,...)}(H), which, by (i), ↪ L^{r}(Ω), ∀1 ≤ r < \frac{k}{2(1−ν)}, if N_{x} ≥ 2, N_{θ} ≥ 6, since −k ≤ 2 − k − 2ν < −\frac{k}{2}

if 1 − \frac{k}{4} < ν ≤ 1 and 2 − k − ∑_{j=1}^{6} δ_{j} = 2 − 3ν − k < k if ν > \frac{2}{3}. Thus, for

any F vanishing on H, F^{3} · I_{N_{x},N_{θ}}^{−k,δ⃗_{0}}(H) ↪ W^{2,r}(Ω) if 1 ≤ r < \frac{k}{2(1−ν)}, N_{x} ≥ 3 and N_{θ} ≥ 6. □

We also have

Proposition 2.3. If A ∈ Ψ^{r}(ℝ^{n}) is properly supported, then for any m ∈ ℝ, δ⃗, N_{x} ≥ 2n and N_{θ} ≥ 0,

(2.4)  A : I_{N_{x},N_{θ}}^{m,δ⃗}(H) → I_{N_{x},N_{θ}}^{m+r,δ⃗}(H)

and ||A|| is bounded by a finite number of semi-norms of σ(A) in S_{1,0}^{r}(ℝ^{n}×(ℝ^{n}\0)).

Proof. Write

Af(x) = ∫ e^{i(x−y)·ξ}a(x, y, ξ)f(y)dξdy, a ∈ S^{r}

and

u(y) = ∫ e^{iF (y)θ}b(y, θ)dθ, b ∈ S_{N_{x},N_{θ}}^{m,δ⃗}.

Then

\begin{matrix}Au(x) = & ∫ e^{i[(x−y)·ξ+F (y)θ]}a(x, y, ξ)b(y, θ)dθdξdy \\ = & ∫ e^{iF (x)θ}[∫ e^{i[(x−y)·ξ+(F (y)−F (x))θ]}a(x, y, ξ)b(y, θ)dξdy]dθ \\ = & ∫ e^{iF (x)θ}c(x, θ)dθ. \end{matrix}

Writing

\begin{matrix}F (y) − F (x) = & ∫_{0}^{1} \frac{d}{ds}(F (x + s(y − x))ds \\ = & (∫_{0}^{1} ∇F (x + s(y − x))ds) · (y − x) \\ =^{def} & G(x, y) · (y − x), \end{matrix}

and making the substitution z = y − x, ζ = ξ − θG(x, y), we obtain

c(x, θ) = ∫ e^{−iz·ζ}a(x, x + z, ζ + θG(x, y))b(x + z, θ)dζdz.

An examination of standard proofs of stationary phase, (cf.,[31,Lem.1.1.8]) shows that it suffices for the amplitude to have two derivatives for each dimension in order to obtain the estimate |c(x, θ)| ≤ C(1 + |θ|)^{m+r}. Noting that θ plays the role of a parameter here, since the stationary phase is in z, ζ, one sees that the derivatives in x and θ satisfy Def. 2.1(i) for order m + r. One concludes that c ∈ S_{N_{x},N_{θ}}^{m+r,δ⃗}. □

To obtain the boundedness of M_{q} on some of the finite-regularity conormal spaces, we will need the following elementary result, which we state without proof.

Lemma 2.4. Let a, b ∈ C(ℝ^{k}) satisfy

(2.5)  |a(θ)| ≤ C(1 + |θ|)^{m} and |b(θ)| ≤ C(1 + |θ|)^{m′}

with m + m′ < −k. Then the convolution a ∗ b satisfies

(2.6)  |a ∗ b(θ)| ≤ C(1 + |θ|)^{m′′},

with m′′ = max((m+k) ˜++m′, m+(m′+k) ˜+, m+m′+k). Here, t ˜+ = t_{+} = max(t, 0) if t ≠ 0 and t ˜+ = ϵ for t = 0, with ϵ > 0 arbitrarily small.

Using this, we show

Proposition 2.5. Let H ⊂ ℝ^{N} be codimension k. For q ∈ I^{μ}(H), let M_{q} denote multiplication by q(x). Suppose \frac{2}{3} − k ≤ μ < 1 − k. Then, for any N_{x} ∈ ℕ,

(2.7)  M_{q} : I_{N_{x},6}^{−k,δ⃗_{0}}(H) → I_{N_{x},6}^{μ̃,δ⃗_{0}}(H) continuously

where, if we write μ = ν − k with \frac{2}{3} ≤ ν < 1,

(2.8)  δ⃗_{0} = (ν, 1 − ν, 2ν − 1, 1 − ν, 1 − ν, 3ν − 2, …) and μ̃ = μ + ϵ

for any 0 < ϵ < −μ.

Remark. Since N_{θ} = 6, we only need specify δ_{1}, …, δ_{6}. Note also that (2.8) is compatible with the δ⃗_{0} in Prop. 2.2(iii).

Proof. As in the proof of Prop. 2.2, we can assume that H = {x′′ = 0},

q(x) = ∫_{ℝ^{k}} e^{ix′′·θ}a(x, θ)dθ, a ∈ S_{1,0}^{μ}

and

u(x) = ∫_{ℝ^{k}} e^{ix′′·θ}b(x, θ)dθ, b ∈ S_{N_{x},6}^{−k,δ⃗_{0}}.

Hence, M_{q}u(x) = ∫ e^{ix′′·θ}(a ∗ b)(x, θ)dθ, where ∗ denotes the k-dimensional convolution in the θ variable. By Lemma 2.4,

|a ∗ b(x, θ)| ≤ C(1 + |θ|)^{max(μ,μ+ϵ,μ)} = C(1 + |θ|)^{μ+ϵ}, ∀ϵ > 0.

Since ∂_{θ}a ∈ S^{μ−1} and μ − 1 < −k,

|∂_{θ}(a ∗ b)(x, θ)| = |(∂_{θ}a) ∗ b(x, θ)| ≤ C(1 + |θ|)^{max(−k,μ−1+ϵ,μ−1)} = C(1 + |θ|)^{−k}

(for ϵ < −μ), which gives a gain of ≥ μ − (−k) = ν, consistent with δ_{1} = ν. (The additional gain of ϵ we choose to ignore.) Noting that ∂_{θ}b ∈ S_{N_{x},3}^{−k−ν,(1−ν,2ν−1,...)} by Remark 4 above, we have

|∂_{θ}^{2}(a ∗ b)| = |(∂_{θ}a) ∗ (∂_{θ}b)| ≤ C(1 + |θ|)^{max(−k−ν,ν−k−1,−k−1)} = C(1 + |θ|)^{ν−k−1}

since ν ≥ \frac{1}{2}, which is a gain of δ_{2} = −k − (ν − k − 1) = 1 − ν. Since ∂_{θ}^{2}a ∈ S^{μ−2} and ∂_{θ}b is as noted above,

|∂_{θ}^{3}(a ∗ b)| = |(∂_{θ}^{2}a) ∗ (∂_{θ}b)| ≤ C(1 + |θ|)^{max(−k−ν,ν−k−2)} = C(1 + |θ|)^{−k−ν},

which gives a gain of δ_{3} = ν − k − 1 − (−k − ν) = 2ν − 1. Since ∂_{θ}^{2}a ∈ S^{μ−2} and ∂_{θ}^{2}b ∈ S_{N_{x},4}^{−k−1,(2ν−1,1−ν,...)},

|∂_{θ}^{4}(a ∗ b)| = |(∂_{θ}^{2}a) ∗ (∂_{θ}^{2}b)| ≤ C(1 + |θ|)^{max(−k−1,ν−k−2,2ν−k−3)} = C(1 + |θ|)^{−k−1},

which gives a gain of δ_{4} = −k − ν − (−k − 1) = 1 − ν. Continuing in this fashion, we may estimate

|∂_{θ}^{5}(a ∗ b)| = |(∂_{θ}^{2}a) ∗ (∂_{θ}^{3}b)| ≤ C(1 + |θ|)^{ν−k−2} ⟹ δ_{5} = 1 − ν if ν ≥ \frac{2}{3},

and

|∂_{θ}^{6}(a ∗ b)| = |(∂_{θ}^{3}a) ∗ (∂_{θ}^{3}b)| ≤ C(1 + |θ|)^{−2ν−k} ⟹ δ_{6} = 3ν − 2.

The x derivatives, not involving any gain in |θ|, are handled in the obvious way. Hence, a ∗ b ∈ S_{N_{x},6}^{μ+ϵ,δ⃗_{0}}. □

Now recall some facts concerning the Faddeev Green’s function [10], G_{ρ}. As is well-known (see,e.g., [32], where this is used implicitly), the families {|ρ|G_{ρ} : ρ · ρ = 0} and {G_{ρ} : ρ · ρ = 0} are uniformly bounded in Ψ^{0}(K) and Ψ^{−1}(K), respectively, for K ⊂⊂ ℝ^{n} and hence interpolation implies

(2.9)  {|ρ|^{1−t}G_{ρ} : ρ · ρ = 0} ⊂ Ψ^{−t}(K) is bounded , ∀t ∈ [0, 1].

We can now state a local analogue for the finite-regularity conormal spaces of the result of [32] concerning solvability of inhomogeneous equations involving Δ_{ρ} +q(x) in weighted L^{2} spaces.

Proposition 2.6. If q ∈ I^{μ}(H) with \frac{2}{3} − k ≤ μ < 1 − k and 0 ≤ σ ≤ 1, then for δ⃗_{0} as in (2.8), the inhomogeneous equation

(2.10)  (Δ_{ρ} + q)w = g ∈ I_{2n,6}^{1−k−σ,δ⃗_{0}}(H)

has, for |ρ| large, a unique solution w ∈ I_{2n,6}^{−k,δ⃗_{0}}(H), with ∥w∥ ≤ \frac{C}{|ρ|^{σ}} ∥g∥.

Proof. Applying G_{ρ} to both sides of (2.10), using (2.9) for t = 1 − σ and Prop. 2.3, we are reduced to showing that

(I + G_{ρ}M_{q})w = G_{ρ}g ∈ I_{2n,6}^{−k,δ⃗_{0}}(H)

has a unique solution for |ρ| sufficiently large, with ∥w∥ ≤ C∥G_{ρ}g∥. By Prop. 2.5, M_{q} : I_{2n,6}^{−k,δ⃗_{0}}(H) → I_{2n,6}^{μ+ϵ,δ⃗_{0}}(H) for any 0 < ϵ < −μ. Note that t = μ + ϵ − (−k) < 1, so we can use Prop. 2.5 and (2.9) with this value of t to obtain

I_{2n,6}^{−k,δ⃗_{0}}(H) →^{M_{q}} I_{2n,6}^{μ+ϵ,δ⃗_{0}}(H) →^{G_{ρ}} I_{2n,6}^{−k,δ⃗_{0}}(H)

with norm ≤ C(q)|ρ|^{t−1} → 0 as |ρ| → ∞. Hence, for |ρ| sufficiently large, ||G_{ρ}M_{q}|| < \frac{1}{2} and I + G_{ρ}M_{q} is invertible on I_{2n,6}^{−k,δ⃗_{0}}(H). □

We may now complete the proof of Thm. 1 as described in §1. Construct two solutions ψ_{j}, j = 1, 2, to (1.10) for potentials q_{j} ∈ I^{μ_{j}}(H_{j}). Note that −q_{j},

the right hand side of (1.10), belongs to I^{μ_{j}}(H) ↪ I_{2n,6}^{1−k_{j}−σ_{j},δ⃗}(H), with σ_{j} > 0 since μ_{j} < 1 − k_{j}. Thus, we may apply Prop. 2.6 with g = −q_{j}, j = 1, 2, and then form as above the corresponding solutions, v_{1}(x, ρ_{1}) = e^{ρ_{1}·x}(1 + ψ_{1}(x, ρ_{1})) of (Δ + q_{1})v_{1} = 0 and w_{2}(x, ρ_{2}) = e^{ρ_{2}·x}(1 + ψ_{2}(x, ρ_{2})) of (Δ + q_{2})w_{2} = 0, with ∥ψ_{j}∥_{I_{2n,6}^{−k_{j} ,δ⃗}} ≤ C|ρ|^{−σ_{j}}. The solutions v_{1} and w_{2} belong to X^{p,r}(Ω), with p, r as

in (1.3). In fact, each is in L^{p}(Ω), and in W^{2,r}(Ω) away from H by Prop. 2.2(i)

and (iii), resp., since r < \frac{1}{2(1−ν)}. Furthermore, since q_{1} ∈ L^{\frac{k_{1}}{ν_{1}} −ϵ}, ∀ϵ > 0, we have

Δv_{1} = −q_{1}v_{1} ∈ (L^{\frac{k_{1}}{ν_{1}} −ϵ}) × (L^{s}), ∀ϵ > 0, ∀s < ∞. Since p > \frac{k_{1}}{k_{1}−ν_{1}} ⟹ p′ < \frac{k_{1}}{ν_{1}} ,

we thus have Δv_{1} ∈ L^{p′}(Ω), and similarly for w_{2}. These solutions are constructed for all large |ρ|. Since n ≥ 3, for any ξ ∈ ℝ^{n}\0 and λ ≥ c|ξ|, one can find ρ_{1}, ρ_{2} ∈ {ρ · ρ = 0} with |ρ_{1}| ≃ |ρ_{2}| ≥ λ and ρ_{1} + ρ_{2} = −iξ.

By the assumption that CD_{q_{2}} = CD_{q_{1}}, there exists a v_{2} ∈ X^{p,r}(Ω) such that (1.7) holds. Applying (1.8) and (1.9), it suffices to show that

(2.11)  ∫_{Ω} e^{−iξ·x}(q_{1} − q_{2})(ψ_{1} + ψ_{2} + ψ_{1}ψ_{2})dx → 0 as λ → ∞.

Since each q_{j} ∈ L^{\frac{k_{j}}{ν_{j}} −ϵ}(Ω), ∀ϵ > 0, and, by (1.3), p > (\frac{k_{j}}{ν_{j}})′, j = 1, 2, we have

∫_{Ω} |q_{1} − q_{2}| · |ψ_{j}|dx ≤ \frac{c}{λ^{ϵ_{j}}} , ϵ_{j} > 0

10 as λ → ∞ by Hölder’s inequality. In fact, p > 2(\frac{k_{j}}{ν_{j}})′ by (1.3), so we have

∫_{Ω} |q_{1} − q_{2}| · |ψ_{1}| · |ψ_{2}|dx ≤ \frac{c}{λ^{ϵ_{12}}} , ϵ_{12} > 0,

as well. Thus, we have shown that CD_{q_{1}} = CD_{q_{2}} implies that { \atop q_{1} − q_{2}}(ξ) = 0, ∀ξ, and hence q_{1} = q_{2}, concluding the proof of Thm.1.

3. Reconstruction of the potential

In this section we prove that the potential q can be obtained constructively from the Cauchy data of Δ + q. We follow here the general method of [25]; see also [26] and [19]. However, there are some additional difficulties in our case because we deal with the set of Cauchy data instead of the Dirichlet-to-Neumann map. Moreover, we work with more complicated function spaces due to the singularities of the potential. We will show:

Theorem 2. Suppose that Ω is Lipschitz and H ⊂⊂ Ω is a submanifold of codimension k. Suppose further that q ∈ I^{μ}(H) is a real potential with ν_{0}(k)−k < μ < 1−k and supp(q) ⊂⊂ Ω. Let p, r satisfy 2 ≤ r < r_{0}(k, ν) and max(p_{0}(k, ν), −\frac{k}{μ}) < p < ∞. Then q can be reconstructed on Ω from the Cauchy data CD_{q} of Δ + q on X^{p,r}(Ω).

To start the discussion of reconstruction, we first show how to obtain a global analogue of Prop. 2.6. For s ∈ ℝ and −1 < δ < 0, let W_{δ}^{s,2}(ℝ^{n}) be the weighted Sobolev space denoted by H_{δ}^{s} in [32]. In [32], it is shown that for 0 ≤ t ≤ 1,

(3.1)  ||G_{ρ}f||_{W_{δ}^{s+t,2}} ≤ \frac{C}{|ρ|^{1−t}} ||f||_{W_{δ+1}^{s,2}}.

Now, let supp(q) ⊂ Ω′ ⊂⊂ Ω and let χ_{0} + χ_{∞} ≡ 1 be a partition of unity subordinate to the open cover Ω ∪ (Ω′)^{c} = ℝ^{n}. Fix \frac{2}{3} − k < μ < 1 − k and let δ⃗_{0} be as in (2.8). For m ≤ 1 − k and s ≤ −2, define

(3.2)  ||f||_{Y_{δ}^{m,s}(ℝ^{n})} = ||χ_{0} · f||_{I_{2n,6}^{m,δ⃗_{0}} (H)} + ||χ_{∞} · f||_{W_{δ}^{s,2}(ℝ^{n})}.

For m ≤ −k, elements of I^{m}(H) are in L_{loc}^{p}(ℝ^{n}), ∀p < ∞, and hence I_{comp}^{m}(H) ↪ W_{δ}^{s,2}(ℝ^{n}), ∀m ≤ 1 − k, s ≤ −1. Combining (3.1) with (2.9) and Prop. 2.3, we have, for 0 ≤ t ≤ 1, (3.3)

||G_{ρ}f||_{Y_{δ}^{m−t,s+t}(ℝ^{n})} ≤ \frac{C}{|ρ|^{1−t}} ||f||_{Y_{δ+1}^{m,s}(ℝ^{n})}, m ≤ 1 − k − t, s ≤ −1 − t, −1 < δ < 0.

Finally, since supp(q) ⊂⊂ Ω, it follows from (2.7) that

(3.4)  M_{q} : Y_{δ}^{−k,s} → Y_{δ}^{μ+ϵ,s}, ∀ϵ > 0.

Arguing as in the proof of Prop. 2.6, but replacing the local finite-regularity spaces with the global spaces Y_{δ}^{m,s}(ℝ^{n}), we obtain the following result:

Proposition 3.1. If q ∈ I^{μ}(H) with \frac{2}{3} − k < μ < 1 − k, s ≤ −2, −1 < δ < 0 and 0 ≤ σ ≤ 1, then the inhomogeneous equation

(3.5)  (Δ_{ρ} + q)w = g ∈ Y_{δ+1}^{1−k−σ,s}(ℝ^{n})

has, for |ρ| large, a unique solution w ∈ Y_{δ}^{−k,s+1−σ}(ℝ^{n}) with ||w|| ≤ \frac{C}{|ρ|^{σ}} ||g||.

Next we will construct the boundary values of the exponentially growing solutions on ∂Ω. For this purpose we use the Green’s function G_{ρ}^{q}(x, y) defined by

(3.6)  (Δ + q)G_{ρ}^{q}(· , y) = δ_{y} in ℝ^{n},
e_{ρ}(·)G_{ρ}^{q}(· , y) ∈ Y_{δ}^{−k,s}(ℝ^{n}),

where y ∈ ℝ^{n} \ \overline{Ω}, e_{ρ}(x) = exp(−ρ· x) and s < −n/2. When |ρ| is large enough, the equation (3.6) has a unique solution by Prop. 3.1. Next we consider the case when ρ is fixed and sufficiently large.

As supp(q) ⊂⊂ Ω, we see that ∂Ω has a neighborhood V such that in V × V Green’s function G_{ρ}^{q}(x, y) has the same singularities as the Green’s function G_{0}^{0} (for the zero potential and ρ = 0), that is, G_{ρ}^{q}(x, y) − G_{0}^{0}(x, y) ∈ C^{∞}(V × V).

Using the Green’s function (3.6) we define the corresponding single and double layer potentials

S_{q}ϕ(y) = ∫_{∂Ω} G_{ρ}^{q}(x, y)ϕ(x)dS_{x}, K_{q}ϕ(y) = ∫_{∂Ω}(\frac{∂}{∂n(x)}G_{ρ}^{q}(x, y))ϕ(x)dS_{x}, y ∉ ∂Ω

which define continuous operators S_{q} : W^{1− \frac{1}{r} ,r}(∂Ω) → X|_{Ω} ⊕ W_{loc}^{2,r}(ℝ^{n} \ Ω) and K_{q} : W^{2− \frac{1}{r} ,r}(∂Ω) → X|_{Ω} ⊕ W_{loc}^{2,r}(ℝ^{n} \ Ω). Here X ⊂ D′(ℝ^{n}) is the space with the norm ||χ_{0} · f||_{I_{2n,6}^{m,δ⃗_{0}} (H)} + ||χ_{∞} · f||_{W_{δ}^{2,r}(ℝ^{n})}. These layer potentials can be considered

as operators on the boundary ∂Ω, defined in principal value sense. Since ∂Ω is Lipschitz, it follows from the results of [7] that these operators are continuous, S_{q}^{∂Ω} : W^{1− \frac{1}{r} ,r}(∂Ω) → W^{2− \frac{1}{r} ,r}(∂Ω) and K_{q}^{∂Ω} : W^{2− \frac{1}{r} ,r}(∂Ω) → W^{2− \frac{1}{r} ,r}(∂Ω). Similarly, on ∂Ω we define normal derivatives of the layer potentials,

∂_{n}S_{q}^{∂Ω}ϕ(y) = p.v. ∫_{∂Ω}(\frac{∂}{∂n(y)}G_{ρ}^{q}(x, y)ϕ(x)dS_{x},

∂_{n}K_{q}^{∂Ω}ϕ(y) = p.v. ∫_{∂Ω}(\frac{∂}{∂n(y)} \frac{∂}{∂n(x)}G_{ρ}^{q}(x, y))ϕ(x)dS_{x}

which are continuous operators ∂_{n}S_{q}^{∂Ω} : W^{1− \frac{1}{r} ,r}(∂Ω) → W^{1− \frac{1}{r} ,r}(∂Ω) and ∂_{n}K_{q}^{∂Ω} : W^{2− \frac{1}{r} ,r}(∂Ω) → W^{1− \frac{1}{r} ,r}(∂Ω).

Next we consider the Calderón projector [5]. We start with the operator

A_{q}(ϕ, ψ) = (−S_{q}^{∂Ω}ϕ + (−\frac{1}{2} + K_{q}^{∂Ω})ψ, −(\frac{1}{2} + ∂_{n}S_{q}^{∂Ω})ϕ + ∂_{n}K_{q}^{∂Ω}ψ)

Proposition 3.2. Let Z = W^{2− \frac{1}{r} ,r}(∂Ω) × W^{1− \frac{1}{r} ,r}(∂Ω). Then the operator

A_{q} : Z/Ker(A_{q}) → Z

is semi-Fredholm. Moreover, −A_{q} : Z → Z is a projection operator with range CD_{q} and kernel independent of q. In particular, this shows that CD_{q} is a closed subspace of Z.

Proof. First we show that kernel of A_{q} does not depend on q. Assume that (ϕ, ψ) ∈ Z. We consider the function

u_{ϕ,ψ}(y) = −S_{q}(ϕ) + K_{q}(ψ), y ∈ ℝ^{n} \ ∂Ω

and the trace operators

T_{+} : W^{2,r}(ℝ^{n} \ Ω) → W^{2− \frac{1}{r} ,r}(∂Ω) × W^{1− \frac{1}{r} ,r}(∂Ω),
T_{−} : W^{2,r}(Ω) → W^{2− \frac{1}{r} ,r}(∂Ω) × W^{1− \frac{1}{r} ,r}(∂Ω)

defined by T_{±}u = (u|_{∂Ω}, ∂_{n}u|_{∂Ω}). As the Green’s functions G_{ρ}^{q}(x, y) have the same singularities near ∂Ω × ∂Ω as the standard Green’s function of ℝ^{n}, we can use the standard jump relations for layer potentials (see e.g. [8]). We conclude that

T_{−}u_{ϕ,ψ} = A_{q}(ϕ, ψ), T_{+}u_{ϕ,ψ} = (ϕ, ψ) + A_{q}(ϕ, ψ).

Thus we get that u = u_{ϕ,ψ} ∈ (e_{ρ})^{−1}Y_{δ}^{−k,s} and it is the unique solution of

(3.7)  (Δ + q)u = g_{ϕ,ψ} = ψδ_{∂Ω} + ∇· (nϕδ_{∂Ω}) in ℝ^{n},

satisfying e_{ρ}(·)u ∈ Y_{δ}^{−k,s}(ℝ^{n}).

Now, if (ϕ, ψ) ∈ Ker(A_{q}) we have that

(ϕ, ψ) = (ϕ, ψ) + T_{−}u_{ϕ,ψ} = T_{+}u_{ϕ,ψ}.

Thus, v = u_{ϕ,ψ} is the solution of the scattering problem

(3.8)  Δv = 0 in ℝ^{n} \ Ω,
T_{+}v = (ϕ, ψ), e_{ρ}(·)v ∈ W_{δ}^{s,2}(ℝ^{n} \ Ω).

On other hand, assume that (3.8) has a solution, and let v_{0} be the zero-continuation of v, that is v_{0}|_{ℝ^{n}\Ω} = v, v|_{Ω} = 0. Then we conclude that v_{0} is a solution of the problem (3.7), and as this solution is unique, we see that v_{0} = u_{ϕ,ψ}. This shows that (ϕ, ψ) ∈ Ker(A_{q}) if and only if the problem (3.8) has a solution. This is obviously independent of q and thus we see that

(3.9)  Ker(A_{q}) = Ker(A_{0}).

Next we consider range of A_{q}. A standard application of Green’s formula (see, e.g., [8, Th. 3.1]) shows that if v ∈ X^{p,r} satisfies

(Δ + q)v = 0 in Ω,

and (ϕ, ψ) = T_{−}v, then v = −u_{ϕ,ψ}. (Observe the negative sign which is due to the fact that we use exterior normal vector n.) Also, by for (ϕ, ψ) ∈ Z we have χ_{0}u_{ϕ,ψ} ∈ I_{2n,6}^{−k,δ⃗_{0}}(H) ⊂ L^{t_{1}}(Ω) for any t_{1} < ∞ by Prop. 2.2. As q ∈ I^{μ}(H) ⊂ L^{t_{2}}(Ω) for 1 < t_{2} < \frac{k}{k+μ} we have χ_{0}Δu_{ϕ,ψ} = −χ_{0}qu_{ϕ,ψ} ∈ L^{p′}(Ω) for \frac{1}{p} < 1 − \frac{1}{t_{2}} , i.e. p > −\frac{k}{μ}. Hence u_{ϕ,ψ} ∈ X^{p,r}. Thus the set of all solutions of the Schrödinger equation in X^{p,r} equals to the set of solutions u_{ϕ,ψ}, (ϕ, ψ) ∈ Z. As T_{−}u_{ϕ,ψ} = A_{q}(ϕ, ψ), we obtain that the range of A_{q} equals to CD_{q}.

Now, when the potential is equal to zero, the Dirichlet-to-Neumann operator Λ_{0} : u|_{∂Ω} = ∂_{n}u|_{∂Ω} is well defined, Λ_{0} : W^{2− \frac{1}{r} ,r}(∂Ω) → W^{1− \frac{1}{r} ,r}(∂Ω). The Cauchy data CD_{0} is the graph of the operator Λ_{0} and is thus closed. Thus we see that the range of A_{0} is a closed subspace, and therefore the operator

A_{0} : Z/Ker(A_{0}) → Z

has zero kernel and closed range. Thus it is a semi-Fredholm operator. Now, consider the operator A_{q} − A_{0}. Using (3.9) we know that the operator

A_{q} − A_{0} : Z/Ker(A_{0}) → Z

is well defined and compact. As compact perturbations of semi-Fredholm operators are also semi-Fredholm, we conclude that A_{q} is also semi-Fredholm.

It remains to show that −A_{q} is a projection. This can be seen similarly to the smooth case. Indeed, if (ϕ, ψ) ∈ Ran(A_{q}), (ϕ, ψ) = A_{q}(ϕ̃, ψ̃) we see that the solution u_{ϕ̃,ψ̃} has the trace T_{−}u_{ϕ̃,ψ̃} = (ϕ, ψ). Hence Green’s formula gives

u_{ϕ̃,ψ̃} = −(−S_{q}ϕ + K_{q}ψ) in Ω.

Taking trace T_{−} from both sides we obtain that (ϕ, ψ) = −A_{q}(ϕ, ψ), i.e. (−A_{q})^{2} = −A_{q}. Thus, Prop. 3.2 is proven. □

Now we can construct the boundary values of the exponentially growing solutions from the Cauchy data. As we are given CD_{q} = Ran(A_{q}), and we know Ker(A_{q}) = Ker(A_{0}), we can construct the projection −A_{q}. Next, let (ϕ_{ρ}, ψ_{ρ}) = T_{+}e_{ρ} be the boundary values of the incoming plane wave. Consider the solution v(x) = e^{ρ·x}(1 + ψ(x, ρ)) = e^{ρ·x} + u_{0} and let (ϕ, ψ) = T_{+}u_{0}. Then (ϕ, ψ) ∈ Ker(A_{q}) and u_{0} = u_{ϕ,ψ} in ℝ^{n} \ Ω. Moreover, as v is solution of Schrödinger equation inside Ω, we have

(3.9)  T_{+}(e_{ρ} + u_{ϕ,ψ}) = (ϕ_{0} + ϕ, ψ_{0} + ψ) + A_{q}(ϕ, ψ) ∈ Ran(A_{q}).

By applying with projection I + A_{q} to (3.9) and using A_{q}(ϕ, ψ) = 0, we see that

0 = (1 + A_{q})(ϕ_{0} + ϕ, ψ_{0} + ψ) = (1 + A_{q})(ϕ_{0}, ψ_{0}) + (ϕ, ψ).

As A_{q} and (ϕ_{0}, ψ_{0}) are known, we find (ϕ, ψ) and the Cauchy data of v(x) on ∂Ω.

So far, we have constructed the Cauchy data of the solutions v_{ρ_{1}}(x) = e^{ρ_{1}·x}(1 + ψ(x, ρ_{1})) for all sufficiently large ρ_{1}. Thus if we consider complex frequencies ρ_{1} and ρ_{2} satisfying ρ_{1} + ρ_{2} = −iξ, with ξ ∈ ℝ^{n}\0, an application of Green’s formula yields

q̂(ξ) = lim_{|ρ_{1}|→∞} ∫_{Ω} q(x)e^{ρ_{1}·x}(1 + ψ_{1}(x, ρ_{1}))· e^{ρ_{2}·x} dx
= lim_{|ρ_{1}|→∞} ∫_{∂Ω}(v_{ρ_{1}}· ∂_{n}e^{ρ_{2}·x} − ∂_{n}v_{ρ_{1}}· e^{ρ_{2}·x}) dx

This proves Theorem 2. □

4. Non-uniqueness for highly singular potentials

We next discuss how very strong singularities of the potential can cause nonuniqueness in a closely related inverse problem. Due to the strength of the singularities, the Schrödinger equation has to be interpreted in a weak sense. Let us consider the boundary value problem

(4.1)  (Δ + q + E)u = 0 in Ω,
u|_{∂Ω} = f

with the potential q having the form

(4.2)  q(x) = −dist (x, H)^{μ}c_{0}(x),

near H, where dist is the Euclidean distance, H is a closed hypersurface bounding a region Ω_{0} ⊂⊂ Ω, μ < −2, and c_{0}(x) is a smooth function, satisfying

c_{0}(x) > C_{0} > 0

in some neighborhood V of H.

Elements of I^{−1−μ}(H) satisfy the pointwise estimate |q(x)| ≤ Cdist (x, H)^{μ}, but a q satisfying (4.2) is not even locally integrable and thus need not define a distribution. Hence, the solutions of (4.1) cannot be formulated in the usual sense of distributions. Instead, we define the solution of (4.1) (if it exists) to be the solution of the following convex minimization problem: Find u such that

(4.3)  G(u) = inf G(v)

where G = G_{q+E} : {v ∈ H^{1}(Ω) : v|_{∂Ω} = f} → ℝ ∪ {∞} is the convex functional

G_{q+E}(v) = ∫_{Ω}(|∇v(x)|^{2} − (q(x) + E)|v(x)|^{2})dx

Here, since the function −q(x) is bounded from below, we define G(u) = ∞ when q|v|^{2} is not in L^{1}(Ω).

Proposition 4.1. The Cauchy data

CD_{q+E} = \{(u|_{∂Ω}, \frac{∂u}{∂n}|_{∂Ω}) : u ∈ H^{1}(Ω), u is a minimizer of G_{q+E}\}.

does not depend on q in Ω_{0}. In particular, if the solution of (4.3) is unique, u vanishes identically in Ω_{0}.

Remark. We note that potentials having singularities similar to (4.2) as above has been used to produce counterexamples to strong unique continuation, e.g. potentials q(x) = c/|x|^{2+ε} in [11]. Recently, counterexamples have been found for weak unique continuation for L^{1}-potentials [18], but here we need to construct potentials for which all solutions vanish inside H. Finally, we wish to emphasize that since the solutions of (4.1) considered here are not defined in the usual sense of distributions, but rather as solutions of a convex minimization problem, the solutions we construct do not give new counterexamples for the unique continuation problem.

Proof. Obviously we can assume that q(x) ≤ 0 everywhere. We start first with the case where E = 0 and f ∈ C^{∞}(∂Ω).

As the potential q is not in the Kato class ([6,p.62]), consider instead a decreasing sequence of smooth functions q_{n} ∈ C^{∞}(Ω), q_{n+1}(x) ≤ q_{n}(x), for which q_{n}(x) = q(x) when d(x, H) > \frac{1}{n} and in some neighborhood V of H

(4.4)  q_{n}(x) ≤ max(−c_{1}n^{−μ}, q(x))

where 0 < c_{1} < C_{0}. Let G_{n} be the functionals defined as G with q replaced with q_{n}. The functionals G_{n} have unique minimizers u_{n} which satisfy in classical sense

(4.5)  (Δ + q_{n})u_{n} = 0 in Ω,
u_{n}|_{∂Ω} = f.

Now, let f ∈ C^{∞}(∂Ω) be fixed. Let F ∈ H^{1}(Ω) be a function for which F |_{∂Ω} = f and F = 0 in some neighborhood of H. By definition of the potentials q_{n}, for sufficiently large n_{0} we have G(F) = G_{n}(F) = G_{n_{0}}(F) for n ≥ n_{0}. Thus for the minimizers u_{n} of G_{n} we have

G_{n}(u_{n}) ≤ C = G_{n_{0}}(F).

Next, by choosing a subsequence, we can assume that the sequences ∫ |∇u_{n}(x)|^{2}dx and ∫ (−q_{n}(x))|u_{n}(x)|^{2}dx are decreasing when n → ∞. Next, let us denote by C_{1}, C_{2} ≤ C the constants

C_{1} = lim_{n→∞} ∫_{Ω} |∇u_{n}(x)|^{2}dx, C_{2} = lim_{n→∞} ∫_{Ω}(−q_{n}(x))|u_{n}(x)|^{2}dx = C_{2}.

Now, we see that u_{n} are uniformly bounded in H^{1}(Ω) and thus by choosing a subsequence we can assume that there is ũ ∈ H^{1}(Ω) such that u_{n} → ũ weakly in H^{1}(Ω). Moreover,

(4.6)  ∫_{Ω} |∇ũ(x)|^{2}dx ≤ C_{1}

As compact operators map weakly converging sequences to strongly converging ones, we have the norm-convergence u_{n} → ũ in L^{2}(Ω). Thus for n_{0} > 0 we have

− ∫_{Ω} q_{n_{0}}(x)|ũ(x)|^{2}dx = lim_{n→∞} ∫_{Ω}(−q_{n_{0}}(x))|u_{n}(x)|^{2}dx ≤

≤ lim_{n→∞} ∫_{Ω}(−q_{n}(x))|u_{n}(x)|^{2}dx ≤ C_{2}.

Since this is valid for any n_{0} we have by monotone convergence theorem

(4.7)  − ∫_{Ω} q(x)|ũ(x)|^{2}dx ≤ C_{2}.

As G ≤ G_{n},

inf G(u) ≤ lim_{n→∞} min G_{n}(u) = C_{1} + C_{2}.

Now, by (4.6) and (4.7) we have

G(ũ) ≤ C_{1} + C_{2}

and thus ũ is a minimizer of G. Now, as G = G_{n} + (G − G_{n}) where G_{n} is a strictly convex functional and G − G_{n} is a convex functional, G is strictly convex. Thus the minimizer is unique. Hence we see that, for every f, the solution ũ of the minimization problem (4.3) exists, is unique, and is given as the L^{2}-limit of the functions u_{n}. We note that the above analysis was based to the fact that the minimization problems for the G_{n} epi-converge to the minimization problem for G [30].

Recalling that Ω_{0} is the region bounded by H, consider functions u_{n} restricted to Ω_{0}. Let t ↦ B_{t} be the Brownian motion in ℝ^{n} starting from x at time t = 0, i.e., B_{0} = x. As the q_{n} are strictly negative smooth functions, they are in the Kato class and the pair (Ω, q_{n}) is gaugeable (see [6], sect. 4.3 and Th. 4.19). By [6], Th. 4.7, the solution u_{n} can be represented by the Feynman-Kac formula

u_{n}(x) = E(exp (∫_{0}^{τ} q_{n}(B_{t})dt) f(B_{τ}))

where τ = τ_{∂Ω} is the first time when the process hits the boundary, i.e., B_{t} ∈ ∂Ω. Here, we assume B_{t} is a version of Brownian motion for which all realizations are continuous curves (see [21] or [6], Th. 1.4). If x ∈ Ω_{0}, the realizations of Brownian motion have to hit H prior to hitting ∂Ω. Denote the first hitting time for H by τ_{H}; thus the first hitting point is B_{τ_{H}}, and τ_{H} < τ_{∂Ω}. (The stopping time τ_{H} is measurable function in the probability space, see [6,Prop. 1.15]).

Let us now denote by p(ρ, η) the probability that the Brownian motion sent from origin at time t = 0 leaves the origin centered ball with radius ρ before time η. Because of the scale-invariance of Brownian motion, p(sρ, s^{2}η) = p(ρ, η) for s > 0. (Indeed, let us consider reparametrized Brownian motion B̃_{t} = sB_{s^{−2}t}. As the probability densities of (B̃_{t_{1}}, B̃_{t_{2}}, …, B̃_{t_{m}}) coincide to those of (B_{t_{1}}, B_{t_{2}}, …, B_{t_{m}}) we see that we see that B̃_{t} is Brownian motion, too.)

Let A_{ρ,η} = {|B_{t} − B_{τ_{H}} | < ρ for τ_{H} ≤ t < τ_{H} + η}. This set is measurable in the probability space and the probability of A_{ρ,η} is P (A_{ρ,η}) = 1 − p(ρ, η).

Let m > 1 and η = η(m) be such that p(1, η) ≥ \frac{m−1}{m}. Now, q is non-positive and by (4.4) q_{n}(x) < max(−c_{1}n^{−μ}, q(x)) in some neighborhood V of H. When s is so small that the s-neighborhood of H is in V, we have by (4.2) that

|E(exp (∫_{0}^{τ} q(B_{t})dt) f(B_{τ}))| ≤E(exp (∫_{τ_{H}}^{τ_{H}+s^{2}η} q(B_{t})dt) ||f||_{L^{∞}})
≤(1 − P (A_{sρ,s^{2}η(m)}))||f||_{L^{∞}}
+P (A_{sρ,s^{2}η(m)})exp( − s^{2}η(m) min(C_{0}s^{μ}, c_{1}n^{−μ}))||f||_{L^{∞}}.

Thus, choosing s = n^{2/μ} we see that for sufficiently large n

(4.8)  ||u_{n}||_{L^{∞}(Ω_{0})} ≤ (\frac{1}{m} + \frac{m − 1}{m}exp (−η(m)c_{1}n^{2(2+μ)/μ}) ||f||_{L^{∞}}.

As u_{n} → ũ in norm in L^{2}(Ω_{0}),

||ũ||_{L^{2}(Ω_{0})} ≤ \frac{1}{m}||f||_{L^{∞}} vol(Ω_{0})^{1/2}.

for any m. Thus we see that ũ = 0 in Ω_{0}.

Next we consider the case when E ∈ ℝ and f ∈ H^{1/2}(∂Ω). First, let H_{r} = {x ∈ Ω : dist (x, H) < r} and let r be so small that q(x) + E < 0 for x ∈ H_{r}. If u is the solution of (4.1) in Ω, then its restriction ũ = u|_{H_{r}} is the solution of boundary value problem

(4.9)  (Δ + q + E)ũ = 0 in H_{r}
ũ|_{∂H_{r}} = f̃,

where f̃ = u|_{∂H_{r}} ∈ C^{∞}(∂H_{r}), that is, ũ is the solution of minimization problem (4.3) in domain H_{r}. Let q_{n} approximate q in H_{r} as above and ũ_{n} be the corresponding solutions of problem (4.9) with q replaced with q_{n}. As above, we see that problem (4.9) is uniquely solvable, ũ_{n} → ũ in L^{2}(H_{r}), and that ũ_{n}(x) can be represented by the Feynman-Kac formula. Let x ∈ Ω_{0} ∩H_{r}, τ̃ be the first time when the Brownian motion sent from x at t = 0 hits ∂H_{r}, and Ã = {B_{t} ∈ Ω_{0} ∩ H_{r} for 0 ≤ t < τ̃}. Let us denote f̃ = f̃_{+} + f̃_{−}, where f̃_{+} vanishes on Ω_{0} ∩ ∂H_{r} and f̃_{−} vanishes on (Ω \ Ω_{0}) ∩ ∂H_{r}. Then we see that

ũ_{n}(x) = P (Ã)E(exp (∫ τ̃ ; 0 q_{n}(B_{t})dt) f̃_{−}(Bτ̃)|Ã)+

(4.10)  +(1 − P (Ã))E(exp (∫ τ̃ ; 0 q_{n}(B_{t})dt) f̃(Bτ̃)|Ã^{c})

where E(·|Ã) is conditional expectation with condition Ã and Ã^{c} denotes the complement of Ã. Note that in the case of Ã^{c}, the process B_{t} hits H at least once. Analyzing how long Brownian motion is near H as above, we see that when n → ∞, the second term on the right hand side of (4.10) goes to zero. Thus ũ(x), for x ∈ H_{r} ∩ Ω_{0}, depends only on q in H_{r} ∩ Ω_{0} and f_{−}. Similarly, we see that ũ(x), x ∈ H_{r} \ \overline{Ω}_{0}, depends only on q in H_{r} \ \overline{Ω}_{0} and f_{+}. Moreover, analogously to (4.8) we see that

lim_{n→∞} ||ũ_{n}|_{H}||_{L^{∞}(H)} = 0.

Choosing a subsequence, we can assume that ũ_{n} → ũ weakly in H^{1}(H_{r}) and thus in norm in H^{3/4}(H_{r}). Hence, by taking the trace H^{3/4}(H_{r}) → L^{2}(H) we see that ũ|_{H} = 0.

In conclusion, for the boundary value problem (4.9) there are well defined maps

T_{+} : f_{+} ↦ ũ|_{H_{r}\\overline{Ω}_{0}} ∈ \{v ∈ H^{1}(H_{r} \ \overline{Ω}_{0}) : v|_{H} = 0\},

T_{−} : f_{−} ↦ ũ|_{H_{r}∩Ω_{0}} ∈ {v ∈ H^{1}(H_{r} ∩ Ω_{0}) : v|_{H} = 0}

where T_{+} depends only on q in H_{r} \ \overline{Ω}_{0} and T_{−} on q in H_{r} ∩ Ω_{0}.

In particular, on the boundaries ∂H_{r} ∩ Ω_{0} and ∂H_{r} ∩ (Ω \ Ω_{0}) we have “independent” Dirichlet-to-Neumann maps

Λ_{+} : f_{+} ↦ ∂_{n}ũ|_{∂H_{r}\\overline{Ω}_{0}}, Λ_{−} : f_{−} ↦ ∂_{n}ũ|_{∂H_{r}∩Ω_{0}},

where n is the exterior normal of H_{r}.

Next, if u is a solution of boundary value problem (4.1) we denote u_{+} = u|_{Ω\\overline{Ω}_{0}} and u_{−} = u|_{Ω_{0}}. To motivate the next step, we observe that u_{+} and u_{−} satisfy “independent” boundary value problems

(Δ + q + E)u_{+} = 0 in Ω \ (H_{r} ∪ Ω_{0})
u_{+}|_{∂H_{r}} = f̃, ∂_{n}u_{+}|_{∂H_{r}\Ω_{0}} = Λ_{+}(u_{+}|_{∂H_{r}\Ω_{0}})

and

(Δ + q + E)u_{−} = 0 in Ω_{0} \ H_{r}
∂_{n}u_{−}|_{∂H_{r}∩Ω_{0}} = Λ_{−}(u_{−}|_{∂H_{r}∩Ω_{0}}).

Now, considering the form of G and the fact that the solution u of boundary value problem (4.1) satisfies u|_{H} = 0, we see that u_{+} = u|_{Ω\\overline{Ω}_{0}} is a minimizer of G in the set \{v ∈ H^{1}(Ω \ \overline{Ω}_{0}) : v|_{∂Ω} = f, v|_{H} = 0\} and u_{−} = u|_{Ω_{0}} is a minimizer of G in the set {v ∈ H^{1}(Ω_{0}) : v|_{∂Ω_{0}} = 0}.

Conversely, if U = v_{+} in Ω \ \overline{Ω}_{0} and U = v_{−} in Ω_{0} where v_{+} and v_{−} are any minimizers of G in the sets \{v ∈ H^{1}(Ω \ \overline{Ω}_{0}) : v|_{∂Ω} = f, v|_{H} = 0\} and {v ∈ H^{1}(Ω_{0}) : v|_{∂Ω_{0}} = 0}, respectively, then U is solution of (4.1).

In particular, we see that the Cauchy data of solutions u of (4.1) on ∂Ω are independent of u|_{Ω_{0}} and thus of q inside H. This finishes the proof of Prop. 4.1. As a concluding remark we note that by using the Courant-Hilbert min-max principle, we see that there always are values of E such that minimization problem for v_{−} has non-zero solutions, that is, there are eigenstates U which have vanishing Cauchy data on ∂Ω. □

Physically, this example has the following interpretation: In theory it is possible to construct a potential wall q(x) such that no particles can “tunnel” through it, using an analogy with quantum mechanics. Thus exterior observers can make no conclusions about the existence of objects or structures inside this wall. Moreover, inside H the solution can be in an eigenstate and its Cauchy data vanishes on the boundary of Ω. Thus, making another analogy with quantum mechanics, in this nest the Schrödinger cat could live happily forever.

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Department of Mathematics, University of Rochester, Rochester, NY 14627

Rolf Nevanlinna Institute, P.O. Box 4, 00014 University of Helsinki, Finland

Department of Mathematics, University of Washington, Seattle, WA 98195