Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schr\"odinger operators

Let $A\in\mathrm{Sym}(n\times n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schr\"odinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s:=(-\nabla\cdot(A(x)\nabla))^s$, $s\in (0, 1)$ and $q\in L^\infty$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $\Omega$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s=1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are longstanding problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schr\"odinger operators.


(Communicated by Jun Zou)
Abstract. Let A ∈ Sym(n×n) be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator L s A + q, where L s A := (−∇ · (A(x)∇)) s , s ∈ (0, 1) and q ∈ L ∞ . We are concerned with the simultaneous recovery of q and possibly embedded soft or hard obstacles inside q by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain Ω associated with L s A + q. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential q. If multiple measurements are allowed, then the surrounding potential q can also be uniquely recovered. These are surprising findings since in the local case, namely s = 1, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

Introduction.
1.1. Mathematical setup and statement of the main results. Let Sym(n×n) signify the space of real-valued n × n symmetric matrices for n ≥ 2. Let A(x) = (a ij (x)) n i,j=1 ∈ Sym(n × n), x ∈ R n . Throughout, it is assumed that A satisfies the following uniform ellipticity condition for some γ ∈ (0, 1), a ij (x)ξ i ξ j ≤ γ −1 |ξ| 2 for all ξ, x ∈ R n , and A(x) is C ∞ -smooth for any x ∈ R n . Define L A to be the following PDO (partial differential operator), Let s ∈ (0, 1) be a constant and introduce the following nonlocal PDO, L s A = (−∇ · (A(x)∇) s , whose rigorous definition shall be given in Section 2.
Let Ω and D be two bounded open sets in R n such that D Ω and, R n \Ω and Ω\D are connected. Let q ∈ L ∞ (Ω\D) be a real-valued function. Physically speaking, q and D, respectively, represent a potential and an embedded impenetrable obstacle inside the potential.
Consider the following nonlocal problem associated with q and D, in Ω e := R n \Ω, where u ∈ H s (R n ) is a weak solution of (1.2) with g ∈ H s (R n ) being an exterior Dirichlet data. In (1.2), Bu := u if D is a soft obstacle, and Bu : is not an eigenvalue of the operator L s A + q, in the following sense in Ω e , and Bw = 0 in D, then w ≡ 0.
Throughout, we assume that {0} is not an eigenvalue of L s A + q, and hence (1.2) is well-posed. In particular, one has the following well-defined Dirichlet-to-Neumann (DtN) map associated to the nonlocal problem (1.2), Λ D,q : H → H * , and Λ D,q is formally given by (1.4) Λ D,q ψ := L s A u ψ | Ωe , where u ψ is the unique solution to (1.2) with u ψ = ψ in Ω e . In the subsequent section, we shall introduce more details of the abstract Banach spaces H and H * . We regard the DtN map Λ D,q as the exterior measurement for our inverse problem study. In this article, we are mainly concerned with the recovery of the embedded obstalce D Ω and the surrounding potential q(x) ∈ L ∞ (Ω\D) by using the exterior DtN map of (L s A + q)u = 0 in Ω\D. For the inverse problem described above, our main results can be stated as follows. ⊂ Ω e be two arbitrary nonempty open sets. Suppose D j and q j ∈ L ∞ (Ω\D j ) satisfy the eigenvalue condition (1.3), j = 1, 2. Let Λ Dj ,qj be the DtN maps for the nonlocal equations (L s A + q j ) u j = 0 in Ω\D j with u j = 0 in D j for j = 1, 2, then the following statements hold.
1. For any given Moreover, if we further assume q(x) = 0 for any x ∈ Ω, then we have the following unique recovery result for the sound hard case.
Let Ω, O j and D j , q j , j = 1, 2, be the same as those described in Theorem 1.1. Let Λ Dj ,qj be the DtN maps for the nonlocal equations (L s A + q j ) u j = 0 in Ω\D j with L s A u j = 0 in D j for j = 1, 2, then the following statements hold. 1. We further assume that q j (x) = 0 for any x ∈ Ω and j = 1, 2. For any given , then one has q 1 = q 2 in Ω\D, where D := D j for j = 1, 2.
Remark 1.1. In this paper, we define the nonlocal Neumann derivative of the solution u to (1.2) as L s A u. This notion is used for defining both the hard obstacle D and the exterior measurement data in (1.4). With such a definition, we show the well-posedness of the direct problem (1.2) as well as derive the unique determination results in Theorems 1.1 and 1.2 for the associated inverse problems. Nevertheless, we would like to point out that there are different ways of introducing the nonlocal Neumann derivative; see for example in [5,Section 3], and the nonlocal Neumann derivative can also be defined by where the integral kernel K s (x, z) is given in (2.1) in what follows. There is the following relationship which connects the aforementioned two definitions of the nonlocal Neumann derivatives (cf. [5, Lemma 3.6]), A u for a different definition of a hard obstacle in (1.2). This leads to a different forward nonlocal problem, and it would be interesting to consider this forward model as well as the associated inverse problems as those in Theorems 1.1 and 1.2. However, this would significantly change our theoretical framework for the present study and we choose to leave it for a future work.
By the first statement in Theorem 1.1 or 1.2, a single pair of non-trivial Cauchy data (g, Λ D,q g) is sufficient to uniquely recover the embedded soft or hard obstacle D, independent of the surrounding potential q. It is also noted that no restrictive regularity assumption is required on the obstacle D. If multiple measurements are used, then both the embedded obstacle and the surrounding potential can be uniquely recovered. We can further show that the recovery of the embedded obstacle can be achieved without knowing whether it is soft or hard. Indeed, by virtue of Theorems 1.1 and 1.2, it suffices for us to establish the following result. Let Ω, O j and D j , q j , j = 1, 2, be the same as those described in Theorem 1.1. Let Λ Dj ,qj be the DtN maps for the nonlocal equations (L s A + q j ) u j = 0 in Ω\D j with either , then the following statements hold.
1. We further assume that q j (x) = 0 for any x ∈ Ω and j = 1, 2. Then for any 1.2. Discussions and historical remarks. The study of nonlocal inverse problems has received significant attention in the literature in recent years. The Calderón problem for the fractional Schrödinger equation was first solved by Ghosh, Salo and Uhlmann [6]. Based on the similar idea, [5] and [14] generalized the results to the nonlocal variable case and nonlocal semilinear case, respectively. Note that the global uniqueness results hold for these nonlocal cases for any space dimension n ≥ 2. The proof of the Calderón problem strongly relies on the strong uniqueness property, and we refer readers to [6, Theorem 1.2] for the fractional Laplace (−∆) s and [5, Theorem 1.2] for the nonlocal variable operator L s A . The strong uniqueness means that: for s ∈ (0, 1), u ∈ H s (R n ), if u = L s A u = 0 in an arbitrary nonempty open set in R n , then u ≡ 0 in R n for any n ≥ 2. Based on the strong uniqueness property, one can obtain the nonlocal Runge approximation property, which states that any L 2 function can be approximated by a sequence of the solutions to (L s A + q)u = 0.
Recently, Rüland and Salo [30] studied the fractional Calderón problem under lower regularity conditions and established the stability results for the determination of unknown potentials. They [29] proved the optimal logarithmic stability for the corresponding inverse problem associated to the fractional Schrödinger equation. In [7], the authors characterized an if-and-only-if relationship between two positive potentials and their associated DtN maps of the fractional Schrödinger equation. Harrach and Lin [7] also provided a reconstruction algorithm of an unknown inclusion based on the monotonicity method. The nonlocal inverse problems reveal some novel and distinctive features compared to their local counterparts. For the current study of simultaneously recovering unknown potentials with possibly embedded impenetrable obstacles, we also provide some interesting discussions and observations compared with its local counterpart.
When s = 1, (1.2) becomes a local problem and in such a case, one should replace the nonlocal condition Bu = 0 in D by Bu = 0 on ∂D, where Bu = u if D is a soft obstacle and Bu = ν T · A · ∇u if D is a hard obstacle, with ν signifying the exterior unit normal vector to ∂D. The corresponding local DtN map can be readily defined on ∂D, which we still denote by Λ D,q . The local inverse problem of determining D by Λ D,q is usually referred to as the obstacle problem. The obstacle problem by a single measurement, namely determining D by a single pair of Cauchy data (ψ, Λ D,q ψ) is a well-known and long-standing problem in the inverse scattering theory, which is also known as the Schiffer's problem, particularly for the case A = I and q = 1 [4,10,21]. There has been extensive study in the literature and significant progress has been achieved in recent years on the Schiffer's problem for the case with general polyhedral obstacles; see [1,3,19,20] and [16,26,27] and the references therein, respectively, for related uniqueness and stability studies. Under the restrictive assumption that ∂D is everywhere non-analytic, the Schiffer's problem was solved in [8]. However, for the case with general obstacles, the Schiffer's problem still remains open in the literature. According to Theorems 1.1-1.3, the nonlocal Schiffer's problem has been completely solved in our study. Hence, it would be much interesting to study the connection of the nonlocal and local Schiffer's problems. This might be partly seen by taking the limit s → 1 − . The simultaneous recovery of an embedded obstacle and an unknown surrounding potential is also a long-standing problem in the literature and closely related to the so-called partial data Calderón problem [9,11]. The existing unique recovery results were established based on knowing the embedded obstacle to recover the unknown potential [9], or knowing the surrounding potential to recover the embedded obstacle [12,13,17,18,24], or using multiple spectral data to recover both of them [15].
The rest of the paper is structured as follows. In Section 2, we provide rigorous mathematical formulations of the nonlocal elliptic operator L s A and fractional Sobolev spaces. In Section 3, we study the well-posedness and the associated DtN map for L s A + q with an embedded obstacle. In Section 4, we prove the uniqueness in determining the obstacle D in Ω by using a single exterior measurement. In Section 5, we prove the global uniqueness in recovering the surrounding potential q. Combining with Section 4 and 5, then we prove Theorem 1.1-1.3.

Preliminary knowledge on L s
A . In this section, we present some preliminary knowledge on the nonlocal PDO L s A that shall be needed in our inverse problem study. We begin with the definition of fractional Sobolev spaces.
2.1. Fractional Sobolev spaces. For 0 < s < 1, the fractional Sobolev space is denoted by H s (R n ) := W s,2 (R n ), which is the standard L 2 based Sobolev space with the norm can also be expressed as and P.V. denotes the standard principal value operator (see [23] for detailed description).
Next, given any open set U of R n and η ∈ R, we consider the following Sobolev spaces, It is known that H η (U ) ⊆ H η 0 (U ), and H η U is a closed subspace of H η (R n ). For more detailed discussions of the fractional Sobolev spaces, we refer to [22,23].

Definition of L s
A . Let us get into the rigorous mathematical formulation for the problem we study here. Let us begin with the definition of the nonlocal operator L s A , s ∈ (0, 1) via the spectral characterization of L A . Suppose that L A is a linear second order self-adjoint elliptic operator, which is densely defined on L 2 (R n ) for n ≥ 2. There is a unique resolution E(λ) of the identity, supported on the spectrum of L A which is a subset of [0, ∞), such that i.e., where dE f,g (λ) is a regular Borel complex measure of bounded variation concentrated on the spectrum of L A , such that d|E f,g |(0, ∞) ≤ f L 2 (R n ) g L 2 (R n ) .
If φ(λ) is a real measurable function defined on [0, ∞), then the operator φ(L A ) is given formally by That is, φ(L A ) is a operator with the domain Following that we define the nonlocal elliptic operators L s A , s ∈ (0, 1) with domain Meanwhile, for any w ∈ H s (R n ), we have For more detailed discussions, we refer readers to [25,28,31].
In fact, the heat-diffusion semigroup admits a nonnegative symmetric heat kernel W t (x, z) for t > 0 by integration, that is for any f ∈ L 2 (R n ) and for any v, w ∈ H s (R n ), which gives the kernel of anisotropic fractional Schrödinger operators and utilizes [2, Theorem 2.4] we have where we use the fact that A(x) is a bounded matrix-valued function defined in R n satisfying (1.1). In addition, the kernel K s possesses the following pointwise estimate (see [2, Theorem 2.4] again) for some constants c 1 , c 2 > 0 depending on A, n and s and K s (x, z) = K s (z, x) for all x, z ∈ R n . We also refer readers to [5] for further discussions of the nonlocal operator L s A .
3. Nonlocal problems with the embedded obstacles and surrounding potentials . In this section, we give the mathematical formulations for our nonlocal problems.
3.1. Well-posedness. In the subsequent discussions, we always let Ω ⊆ R n to be a bounded open set and D Ω to be an open subset, q to be a potential in L ∞ (Ω\D) and s ∈ (0, 1) to be a constant. Consider the nonlocal Dirichlet problem in Ω e .
Define the bilinear form B q (·, ·) by for any v, w ∈ H s (R n ). Combining (2.2) and (3.2), we have that Then by using the standard variational formula, the weak solution can be defined by Let Ω be a bounded open set in R n . Given f ∈ L 2 (Ω\D) and g ∈ H s (R n ), we call that u ∈ H s (R n ) is a (weak) solution to (3.1) provided that u g := u − g ∈ H s (Ω) and Next, we have the following well-posedness. Moreover, when q satisfies the eigenvalue condition (1.3), we have the stability estimate where C > 0 is a constant independent of f and g.

Proof. A straightforward computation shows that
for all w ∈ C ∞ c (Ω\D). It is easy to see that the bilinear form B q (·, ·) is bounded, coercive and continuous by using the pointwise estimate (2.3) of the kernel K s (x, z), then the stability estimate (3.4) follows from the standard Lax-Milgram theorem (a similar proof has been addressed in [5,Section 3]). This completes the proof. We also refer readers to [5,6] for more detailed discussions. Since the solution u ∈ H s (R n ) of (3.1) only depends on the exterior value, in order to simplify notations, we shall consider the Dirichlet data g in the quotient space H in the subsequent studies.
3.2. The DtN map. We define the associated DtN map for L s A +q via the bilinear form B q in (3.3).  (1.3). Let H be the abstract space given in (3.5). Define where u g ∈ H s (R n ) is the solution to (1.2) with the exterior Dirichlet data g. Then Λ D,q : H → H * is a bounded linear map. Moreover, we have the following symmetry property for Λ D,q , Λ D,q g, h H * ×H = Λ D,q h, g H * ×H , g, h ∈ H.
Proof. Combining with Lemma 3.2, the proof is similar to [5, Proposition 3.5], so we skip it here.
Remark 3.1. For any h ∈ H, there exists h ∈ H s (R n ) such that h = h in Ω e . A direct calculation shows that Then from (3.6), we have which implies that Λ q g = L s A u g | Ωe .
The integral identity allows us to solve the nonlocal type inverse problem as a direct consequence of Proposition 3.1. It can be stated as follows. Ω be a obstacle. Let s ∈ (0, 1) and q ∈ L ∞ (Ω\D) satisfy (1.3). For any g 1 , g 2 ∈ H, one haŝ where u j ∈ H s (R n ) solves (L s A + q j )u j = 0 in Ω\D with u j | Ωe = g j for j = 1, 2, and r Ω\D u refers to the restriction of u on Ω\D.
4. Recovery of the obstacle D. In this section, we show that the obstacle D can be uniquely recovered by a single measurement. The following strong uniqueness property shall be needed.
Note that the set Ω \ ∪ 2 i=1 D i is a non-empty open set. From the condition Λ D1,q1 g = Λ D2,q2 g in O 2 and Λ Dj ,qj g = L s A u j | Ωe , one can see that By applying the strong uniqueness property (Proposition 4.1), we obtain w ≡ 0 in R n , which shows u 1 = u 2 in R n .
Second, we claim that D 1 = D 2 in R n by using contradiction arguments. Suppose that D 1 = D 2 . Without loss of generality, we can assume that there exists a nonempty open subset M D 2 \D 1 . Then we have the following two cases.

Case 1.
Either By using the condition u 1 = u 2 in R n , we know that u 1 = u 2 = 0 in M D 2 . Applying the nonlocal elliptic equation for u 1 in M , it is readily seen that L s A u 1 = u 1 = 0 in M. Utilizing the strong uniqueness property again, we obtain that u 1 ≡ 0 in R n .

Case 2.
Either By using the equation of u 1 and q 1 (x) = 0 for x ∈ Ω, we have u 1 = L s A u 1 = 0 in M. Therefore, we have u 1 ≡ 0 in R n by the strong uniqueness property.
However, in either Case 1 or Case 2, the conclusion u 1 ≡ 0 in R n contradicts to the fact that u 1 = g in Ω e with a non-identically zero exterior data g. This proves the first part of Theorem 1.1 to Theorem 1.3 by using a single exterior measurement.
Remark 4.1. Indeed, we do not need to use any information about the solution w in Ω\(D 1 ∪ D 2 ). We only utilize the strong uniqueness of w in the exterior domain Ω e , which is a powerful tool in dealing with the nonlocal type inverse problems.

5.
Recovery of the surrounding potential q. In this section, we prove the uniqueness in determining the surrounding potential q in Ω \ D.
Then A is dense in L 2 (Ω\D).
Proof. The proof follows a similar argument to that of [5,Lemma 5.7]. For the completeness of this paper, we present a detailed proof in what follows. By the Hahn-Banach theorem, it is sufficient to show that for any v ∈ L 2 (Ω\D) with´Ω \D vw dx = 0 for all w ∈ A, then it must satisfy v ≡ 0 in Ω \ D. If v is a such function, which means v satisfies (5.1)ˆΩ \D v · r Ω\D Φ q g dx = 0, for any g ∈ C ∞ c (O).
We claim that where φ ∈ H s (R n ) is the solution given by Lemma 3.1 to . In other words, B q (φ, w) =´Ω \D v ·r Ω\D w dx for any w ∈ H s (Ω\D). To prove (5.2), let g ∈ C ∞ c (O), and we denote u g := Φ q g ∈ H s (R n ) such that u g − g ∈ H s (Ω). Then we havê in which we have used the fact that u g is a solution and φ ∈ H s (Ω\D).
Combining (5.1) and (5.2), we can obtain that B q (φ, g) = 0, for any g ∈ C ∞ c (O). Using the fact that r Ω\D g = 0, since g ∈ C ∞ c (O) we can derive that R n L s A φ · g dx = 0 for any g ∈ C ∞ c (O), and thus we obtain that φ ∈ H s (R n ) satisfies L s A φ| O = φ| O = 0. By the strong uniqueness property again, we know that φ ≡ 0 in R n and also v ≡ 0 in Ω\D. This finishes the proof.
Remark 5.1. It is easy to see that the soft or hard condition Bu = 0 in D does not affect the conclusion of the previous lemma.

5.2.
Proof of the uniqueness in determining q. From the equal DtN maps, by the first statements of Theorems 1.1-1.3, we know that the embedded obstacle D is uniquely recovered. Next, we prove the global uniqueness in determining the potential q ∈ L ∞ (Ω\D). Assume that Λ D,qj are the DtN maps with respect to (L s A + q j )u j = 0 for j = 1, 2. If Λ D,q1 g| O2 = Λ D,q2 g| O2 , for all g ∈ C ∞ c (O 1 ) with u 1 = u 2 = g in Ω e , then one can conclude that q 1 = q 2 in Ω\D.
Proof. Since Λ D,q1 g| O2 = Λ D,q2 g| O2 for all g ∈ C where u j ∈ H s (R n ) are the solutions to (L s A +q j )u j = 0 in Ω\D with the associated exterior values g j ∈ C ∞ c (O 1 ), for j = 1, 2 respectively. Given ϕ ∈ L 2 (Ω\D), by Proposition 5.1, suppose that there exists two sequences (u in Ω e , for some exterior data g → 0 in L 2 (Ω\D) as k → ∞. Substituting these solutions into the integral identity (5.3) and taking the limit as k → ∞, we can deduce that Ω\D (q 1 − q 2 )ϕ dx = 0 Since ϕ ∈ L 2 (Ω\D) is arbitrary, we readily see that q 1 = q 2 in Ω \ D. This also completes the second parts of Theorems 1.1-1.3.