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

Abstract

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.

A practical method for recovering Sturm-Liouville problems from the Weyl function

“…In particular, the Hochstadt-Lieberman problem, the inverse transmission eigenvalue problem, partial inverse problems for quantum graphs. See [36] and references therein for more details.…”

Coincidence Theory of a Nonlinear Periodic Sturm–Liouville System and Its Applications

“…The latest research trends on the S-L equation mainly include the following aspects. The first involves the theoretical and numerical methods and applications of inverse S-L problems (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). The second focuses on the investigation of some generalized S-L equations, such as the fractional differential S-L equation (see [13][14][15][17][18][19][20][21][22][23][24]) and the S-L equation on time scales (see [25][26][27][28][29][30][31][32][33][34]).…”

Determination of a differential pencil from interior spectral data on a union of two closed intervals

“…They proved that the spectrum of the problem −y + q(x)y = λy, t ∈ (0, 1) y (0) − hy(0) = y (1) + Hy(1) = 0 and the logarithmic derivatives of the eigenfunctions at the point 1/2 uniquely determine the potential q(x) on the whole interval [0, 1] almost everywhere. This kind of problems for the differential operators on a continuous interval were studied in [8]- [14], [18,22,23], [28]- [34].…”