Inverse transmission problems for magnetic Schrödinger operators
This paper is concerned with the study of inverse transmission problems for magnetic Schrödinger operators on bounded domains and in all of the Euclidean space, in the self-adjoint case. Assuming that the magnetic and electric potentials are known outside of a transparent obstacle, in the bounded domain case, we show that the obstacle, the transmission coefficients, as well as the magnetic field and electric potential inside the obstacle are uniquely determined from the knowledge of the set of the Cauchy data for the transmission problem, given on an open subset of the boundary of the domain. In the case of the transmission scattering problem, we obtain the same conclusion, when the scattering amplitude at a fixed frequency is known. The problems studied in this work were proposed in .
1. Introduction and statement of results
Let , , be a bounded domain with connected Lipschitz boundary, and let be a bounded open set with Lipschitz boundary such that is connected. Setting also , and letting , , we consider the magnetic Schrödinger operators
where . Let satisfy the magnetic Schrödinger equations
Denote by the almost everywhere defined outer unit normal to and to . Since and , the traces of the normal derivatives and on are well-defined as elements of , see [27, Chapter 3] as well as Subsection 2.1 below. In addition to (1.1), we require that satisfies the following transmission conditions on ,
as well as the Dirichlet boundary conditions on the boundary of ,
Here (the space of –functions with Lipschitz gradient in a neighborhood of ), , and . The transmission conditions (1.2) encompass physical models of imperfect transmission arising in acoustics, elastodynamics, quantum scattering, and semiconductor physics.
In what follows we shall assume that in . Then the Fredholm alternative holds for the transmission problem (1.1), (1.2), and (1.3), see Proposition 3.1 below. Let be an open nonempty subset of the boundary of and let
The first inverse problem studied in this paper is as follows. Assume that we are given the bounded domain , the subset , the magnetic and electric potentials , , and the set of the Cauchy data for the transmission problem (1.1), (1.2), associated with . The problem is whether we can recover the obstacle , the transmission coefficients , , and on , as well as the magnetic and electric potentials , in .
This problem was proposed in , where the corresponding inverse transmission problem in the absence of magnetic potentials was investigated. As it was pointed out in , in general one cannot hope to recover the transmission coefficients , , and on uniquely, since the set of the Cauchy data enjoys the following invariance property,
for any on , constant on each connected component of .
In the presence of the magnetic potentials there is another gauge transformation that preserves the set of the Cauchy data. Namely, for any function , we have
Thus, for any function such that , we have
Notice that the invariance of the set of the Cauchy data under the gauge transformation with is the standard obstruction to the unique determination of the magnetic potential in inverse boundary value problems, see [29, 38].
In general, the transmission problem (1.1), (1.2), and (1.3) is non-self-adjoint. As we shall see in Section 3 below, the self-adjointness of the transmission problem is guaranteed by the assumptions that , are real-valued, and on .
In this paper we shall be concerned with inverse transmission problems in the self-adjoint case. The importance of the self-adjoint transmission conditions comes in particular from the fact that they assure the continuity of the energy flux of the solution of the transmission problem along the boundary of the obstacle , i.e.
Working with self-adjoint transmission problems, the obstruction (1.4) can be eliminated, as shown in the following, first main result of this paper.
Let , , be a bounded domain with connected Lipschitz boundary, and be bounded open subsets with Lipschitz boundaries such that is connected, . Let , , , , , and , . Assume that on , on , and
for an open non-empty subset , then
Furthermore, if is of class , there is a function , , such that
In order to recover the obstacle and the transmission coefficients in the proof of Theorem 1.1, we shall follow closely , and use the method of singular solutions for the transmission problem, with singularities approaching the boundary of the obstacle. The presence of the magnetic potentials complicates the arguments, and we have therefore attempted to give a careful discussion througout. When constructing the singular solutions, it becomes essential to assure that the unique solvability of the transmission problem can always be achieved by a small perturbation of the boundary of a domain. Furthermore, this property is required when establishing some auxiliary Runge type results on approximation of solutions of transmission problems in subdomains by solutions in larger domains, in Subsection 5.2. We show that this key property is enjoyed by the self-adjoint transmission problem in Section 4, through an application of the mini-max principle.
In the second part of Theorem 1.1, we assume that the boundary of is of class . Indeed, to the best of our knowledge, the most general boundary reconstruction result for the tangential component of a continuous magnetic potential from the knowledge of the Dirichlet–to–Neumann map has been obtained in , when the boundary is of class . Next, as far as we know, the most general result, in the sense of regularity, for inverse boundary value problems for the magnetic Schrödinger operator has been proven in  making use of , under the assumption that the boundary of the domain is of class , the magnetic potential is of class , and the electric potential is of class . Here stands for the space of Dini continuous functions, see . We have therefore decided to avoid considering the issue of getting the minimal regularity assumptions on magnetic potentials and on the boundary of , and will content ourselves with Lipschitz continuous magnetic potentials. The minimal regularity of the boundary of required when working with Lipschitz continuous magnetic potentials seems to be . In particular, this is due to the fact that, in general, a solution to the equation on will be of class only when is of class , see [33, Lemma 5.8].
We would like to emphasize that the set of the Cauchy data in Theorem 1.1 can be given on an arbitrarily small open non-empty subset of the boundary of . This is important from the point of view of applications, since in practice, performing measurements on the entire boundary could be either impossible or too cost consuming. To the best of our knowledge, the only available result in the presence of an obstacle, where the measurements are performed on an arbitrarily small portion of the boundary, is the work  for the Schrödinger operator without a magnetic potential. When no obstacle is present and the electric and magnetic potentials are known near the boundary, it is proven in , see also , that the knowledge of the Cauchy data on an arbitrarily small part of the boundary determines uniquely the magnetic field and the electric potential in the entire domain. Dropping the assumption that the potentials are known near the boundary, some fundamental recent progress on inverse boundary value problems with partial measurements has been achieved in [4, 7, 17].
The second part of the paper is devoted to the inverse scattering problem for the magnetic Schrödinger operator in the presence of a transparent obstacle. Here we assume that , , is a bounded open set with Lipschitz boundary such that is connected. Let , , , . Assume that and are compactly supported. As before, let , , be such that in and on .
Let , , and consider the scattering transmission problem,
It is known that the scattered wave has the following asymptotic behavior,
The second inverse problem studied in this paper is as follows. Assume that we are given the scattering amplitude for all and for some fixed , as well as the magnetic and electric potentials and . The problem is whether this information determines the obstacle , the transmission coefficients , , and on , as well as the magnetic and electric potentials , in . In this direction we have the following result, which is a generalization of [15, Theorem 1.2] and .
Let , , be bounded open sets with Lipschitz boundaries such that is connected, . Let , , , . Assume that and are compactly supported. Let , and , . Assume that on , on , and
for all and for some fixed , then
Furthermore, if is of class , there is a function , , such that
The study of inverse obstacle scattering at a fixed frequency has a long tradition, starting with the uniqueness proof for the Dirichlet boundary conditions, going back to Schiffer , see also . Another important technique for the identification of obstacles, applicable to the Neumann and transmission problems, is the method of singular solutions, developed in [11, 13]. Among further important contributions to the circle of questions around inverse transmission obstacle scattering, we should mention [18, 19, 31, 39]. See also the review paper  and the references given there.
The plan of the paper is as follows. After preliminaries in Section 2, the solvability of the transmission problem (1.1), (1.2), and (1.3) in the general non-self-adjoint case is discussed in Section 3 by means of a variational approach, convenient here due to the low regularity of the boundary of the obstacle. In section 4 we show that in the self-adjoint case, the unique solvability of the transmission problem can be achieved by a small perturbation of the boundary of the domain. Section 5 is devoted to the solution of the inverse transmission problem on a bounded domain and to the proof of Theorem 1.1. In the approach to the reconstruction of the obstacle and of the transmission coefficients, following , we use the method of singular solutions for the transmission problem. The task of the reconstruction of the obstacle and of the transmission coefficients occupies Subsections 5.1 – 5.4.
Once the obstacle and the transmission coefficients have been recovered, the determination of the magnetic and electric potentials inside the obstacle becomes possible. This is the subject of Subsection 5.5. Proceeding in the spirit of inverse boundary value problems for the magnetic Schrödinger operator, in order to exploit a fundamental integral identity, valid inside the obstacle, it becomes essential to determine the values of the tangential component of the magnetic potential on the boundary of the obstacle. To this end, in Section 6, we adapt the method of  of the boundary reconstruction of the tangential component of the magnetic potential to our situation, by combining it with an idea of . With the tangential components of the magnetic potential determined, the exploitation of the integral identity becomes possible, and using the machinery developed in [7, 20, 29, 33, 38] for inverse boundary value problems for the magnetic Schrödinger operator, we are able to determine the magnetic and electric potentials inside the obstacle, up to a natural gauge transformation.
Section 7 is concerned with the scattering transmission problem. First, in Subsection 7.1, using the Lax-Phillips method, we investigate the existence of solutions to the scattering transmission problem in the non-self-adjoint case. This discussion generalizes , where the case without magnetic potentials is treated. In Subsection 7.2, the inverse scattering transmission problem is studied, and following the arguments of , we show that Theorem 1.2 is implied by Theorem 1.1.
In Appendix A, we present a unique continuation result for elliptic second order operators from Lipschitz boundaries, which is used several times in the main text. Although this result is essentially well-known, since it plays an important role in the paper, we give it here for the convenience of the reader. Appendix B contains some estimates for fundamental solutions of the magnetic Schrödinger operator, which are crucial when estimating singular solutions of the transmission problems. Finally, in Appendix C we provide a brief discussion of asymptotic bounds on some volume and surface integrals, required in the reconstruction of the obstacle and of the transmission coefficients in the main text.
2.1. Sobolev spaces and traces
Let , , be a bounded open set with Lipschitz boundary. Let , a.e. in , and consider the space
which is a Hilbert space, equipped with the norm
The map is continuous on with values in , see [5, Theorem 1.21]. We have for ,
In this paper, we shall work with a subspace of , which contains , for which the trace of the normal derivative of is still well-defined. To introduce this space, we shall need to consider the –dual of , given by
and , see . Here we use the natural inner product on ,
Notice that in general,
Thus, the restriction does not determine uniquely, and therefore, cannot be imbedded into the space .
The following result will allow us to define the trace of the normal derivatives of functions from a suitable subspace of .
[27, Lemma 4.3] Let be a bounded open set with Lipschitz boundary. Let and satisfy
where , a.e. in . Then there exists such that
for any . Furthermore, is uniquely determined by and , and we have
In what follows we shall write , when the element is given. In particular, when , we shall always make the natural choice
which allows us to recover the standard definition of the trace of a function . Here is the characteristic function of .
In the sequel, we shall also need the following result.
[27, Theorem 3.20] Let be an open subset of . Let for some integer , and let , . If , then and
The same conclusion holds with replaced by .
2.2. Magnetic Schrödinger operators
Let be a bounded open set with Lipschitz boundary, and let and . Let and let satisfy the magnetic Schrödinger equation,
Then we have the following first Green formula,
for any , see (2.2). If and satisfies
then we also have the following first Green formula,
for any . Hence,
In what follows we shall also work with the operator of the form , where , and on . Let and let satisfy,
and using (2.2), we get the following first Green formula, when ,
For any such that , we have the second Green formula,
which is a consequence of (2.7).
Finally, notice that can be extended to the whole of so that the extension, which we denote by the same letter, satisfies . For the existence of such an extension in the case of Lipschitz domain , we refer to [37, Theorem 5, p. 181]. Also we have .
3. Direct transmission problem
Let , , be a bounded open set with Lipschitz boundary, and let be a bounded open subset with Lipschitz boundary. We set as before
Let , , , , , , , and . Notice that since the boundary of is merely Lipschitz, it is convenient to assume here that the transmission coefficients and are defined near , rather than on the boundary . This is due to the fact that the –regularity of is needed in order to eliminate the jump across the interface in the solution of the transmission problem, while still working with second order differential operators with bounded coefficients. The corresponding regularity of the coefficient is needed for similar purposes, when considering the adjoint transmission problem.
Assume furthermore that on . For , we consider the following inhomogeneous transmission problem,
and the corresponding homogeneous transmission problem,
In this paper we shall treat only the Dirichlet boundary conditions on the boundary of , and to this end we introduce the following subspace of ,