On a regularization approach to the inverse transmission eigenvalue problem

Abstract

We consider the irregular (in the Birkhoff and even the Stone sense) transmission eigenvalue problem of the form −y″ + q(x)y = ρ 2 y, y(0) = y(1) cos ρa − y′(1)ρ −1 sin ρa = 0. The main focus is on the ‘most’ irregular case a = 1, which is important for applications. The uniqueness questions of recovering the potential q(x) from transmission eigenvalues were studied comprehensively. Here we investigate the solvability and stability of this inverse problem. For this purpose, we suggest the so-called regularization approach, under which there should first be chosen some regular subclass of eigenvalue problems under consideration, which actually determines the course of the study and even the precise statement of the inverse problem. For definiteness, by assuming q(x) to be a complex-valued function in W 2 1 [ 0 , 1 ] possessing the zero mean value and q(1) ≠ 0, we study properties of transmission eigenvalues and prove the local solvability and stability of recovering q(x) from the spectrum along with the value q(1). In the appendices, we provide some illustrative examples of regular and irregular transmission eigenvalue problems, and we also obtain necessary and sufficient conditions in terms of the characteristic function for the solvability of the inverse problem of recovering an arbitrary real-valued square-integrable potential q(x) from the spectrum for any fixed a ∈ R .

A new approach to the inverse discrete transmission eigenvalue problem

“…Clearly, {µ j } l j=1 and {ν j } l−1 j=1 are the eigenvalues of the boundary value problems for the system (8) with the boundary conditions y 0 = y l+1 = 0 and y 1 = y l+1 = 0, respectively. For {ν j } l−1 j=1 , it is supposed that n = 2, l in (8). In view of ( 13), (14), and Lemma 2.1, the two spectra {µ j } l j=1 and {ν j } l−1 j=1 uniquely specify M (λ), and vice versa.…”

“…The inverse transmission eigenvalue problem consists in reconstruction of the function ρ(x), which is related with the speed of sound, from the eigenvalues of the problem (1). The majority of the studies of the inverse transmission eigenvalue problem (see [19,20,11,1,12,10,7,9,13,24,8]) deal with the radially symmetric case, when the problem (1) is reduced to the one-dimensional form (2)…”

“…Given eigenvalues {λ j } 2l−1 j=1 of the problem ( 8)-( 9), find {a n } l−1 n=1 and {b n } l n=1 . Note that the coefficient a l is multiplied by y l+1 = 0 in (8), so this coefficient cannot be recovered. Although we are primarily interested in Inverse Problem 1.2 in connection with Inverse Problem 1.1, our method for solving Inverse Problem 1.2 is developed for the case of arbitrary relatively prime polynomials R 0 (λ) and R 1 (λ) satisfying some additional conditions.…”

“…We reduce Inverse Problem 1.2 to the reconstruction of {a n } l−1 n=1 and {b n } l n=1 from the Weyl coefficients defined in Section 2. The latter problem is equivalent to the classical inverse problem that consists in the reconstruction of the coefficients from the two spectra {µ j } l j=1 and {ν j } l−1 j=1 of the eigenvalue problems for equations (8) with the boundary conditions y 0 = y l+1 = 0 and y 1 = y l+1 = 0, respectively. In order to deal with this classical inverse problem, we adapt the methods of Yurko [25,26].…”

“…The paper is organized as follows. In Section 2, the solution of the auxiliary inverse problem for equations (8) by the Weyl coefficients is described. In Section 3, we provide our main results concerning Inverse Problems 1.1-1.2.…”