Distribution of transmission eigenvalues and inverse spectral analysis with partial information on the refractive index

Abstract

In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as y 00 .

On the direct and inverse transmission eigenvalue problems for the Schrödinger operator on the half line

“…Therefore, it is possible that nonreal transmission eigenvalues exist. The existence and distribution of the nonreal transmission eigenvalues for the Dirichlet case have been studied in Colton et al, 8,9 Wang and Shieh, 13 and Xu et al 16,17 In the current paper, we show that in some cases, there exist infinitely many real eigenvalues and at most a finite number of nonreal eigenvalues, and in some other cases, there are infinitely many nonreal eigenvalues and at most a finite number of real eigenvalues. Moreover, we give the asymptotic behavior of the transmission eigenvalues (including real and nonreal).…”

“…Consider the following problem R(q, h) ∶ − ′′ + q(x) = , 0 < x < 1, (1.1) (1) = ′ (1), (1.5) where the potential q(x) is real and belongs to L 2 (0, 1), is the spectral parameter, and h ∈ R. The values for which the problem R(q, h) has a pair of nontrivial solutions { , 0 } are called transmission eigenvalues, which are the energies at which the scattering from the "perturbed" system agrees with the scattering from the "unperturbed" system. 1 Recently, the transmission eigenvalue problems have attracted wide attention (see, e.g., Aktosun et al, [1][2][3] Bondarenko and Buterin, 4 Buterin et al, 5,6 Cakoni et al, 7 Colton et al, 8,9 Chen, 10 McLaughlin et al, 11,12 Wang and Shieh, 13 Wei and Xu, 14 Xu et al, [15][16][17][18] Yang, 19 Yang and Buterin 20 and the references therein). There are many interesting studies for the transmission eigenvalue problem with Dirichlet boundary condition at x = 0 (i.e., (0) = 0 = 0 (0) instead of Equations 1.3 and 1.4, respectively).…”

Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem

“…where q(x) is defined in (11). Some aspects of the asymptotics of large (real and non-real) transmission eigenvalues for the case a = 1 were discussed in [19]. In 2015, Colton and co-authors [6] studied the existence and distribution of the non-real transmission eigenvalues.…”

Reconstruction for a class of the inverse transmission eigenvalue problem

“…The numerical experiments are provided to demonstrate the efficiency of our algorithms. Although eigenvalues of L ( q ,1) are complicated for various cases (see Wang and Shieh and Xu et al), the numerical experiments might be almost the same. For simplicity, we assume that all eigenvalues of L ( q ,1) are real‐valued and satisfy .…”

“…The transmission eigenvalue problem L ( η ) is of the form associated with boundary conditions where λ is the spectral parameter and the refractive index η ( r ) is positive and real‐valued, . The inverse transmission eigenvalue problem arose from inverse scattering theory and had been studied by researchers(see previous studies and references therein). The quantity is physically explained as the time for the wave travels from 0 to 1.…”