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

Abstract

This work deals with the interior transmission eigenvalue problem: y +k 2 η (r) y = 0 with boundary conditions y (0) = 0 = y (1) sin k k − y (1) cos k, where the function η(r) is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption for the square of the index of refraction η(r). Moreover, we provide a uniqueness theorem for the case 1 0 η(r)dr > 1, by using all transmission eigenvalues (including their multiplicities) along with a partial information of η(r) on the subinterval. The relationship between the proportion of the needed transmission eigenvalues and the length of the subinterval on the given η(r) is also obtained.

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).…”

“…Proof. The uniqueness has been proved in Xu et al 17 (see also Fedoryuk 27 ). Let us show Equation (2.28).…”

“…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).…”

“…Moreover, we give the asymptotic behavior of the transmission eigenvalues (including real and nonreal). In the Dirichlet case, 17 we gave the asymptotics of the nonreal eigenvalues without the description of the subscript numbers, which will be supplemented in this paper. The asymptotics of (nonreal) eigenvalues with the description of subscript numbers is more difficult to investigate (due to the non-selfadjointness), which has important applications in the inverse spectral problems, especially for the stability and reconstruction algorithm (see, e.g., Bondarenko and Buterin, 4 Bondarenko, 22 Freiling and Yurko, 23 and Rundell and Sacks 24,25 ).…”

An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition

“…Our results also can be applied to this case. The problem (5.2) attracted much attention of both mathematicians and physicists in connection with the inverse acoustic scattering problem (see [6,12,[30][31][32]39] and references therein). In particular, McLaughlin and Polyakov [30] stated the problem of recovering the potential on the interval 0, |a−1|…”