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.

In this work, we study the matrix Sturm-Liouville operator with the separated self-adjoint boundary conditions of general type, in terms of two unitary matrices. Some properties of the eigenvalues and the normalization matrices are given. Uniqueness theorems for determining the potential and the unitary matrices in the boundary conditions from the Weyl matrix, two characteristic matrices or one spectrum and the corresponding normalization matrices are proved.

In this work, we consider the Sturm-Liouville operator on a finite interval [0, 1] with discontinuous conditions at 1/2. We prove that if the potential is known a priori on a subinterval [b, 1] with b ≥ 1/2, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case b < 1/2, parts of either one or two spectra can uniquely determine the potential and a part of parameters.

The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coefficients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction.

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