Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

In this paper, we for the first time get constructive solution for the inverse Sturm–Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl functi...

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Bibliographic Details
Published in:Analysis and mathematical physics Vol. 13; no. 5
Main Authors: Chitorkin, Egor E., Bondarenko, Natalia P.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.10.2023
Subjects:
Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
34L40
Constructive algorithm
Inverse spectral problems
34B24
34A55
Polynomials in the boundary conditions
Sturm–Liouville operator
Uniqueness theorem
34B09
Singular potential
34B07
ISSN:1664-2368, 1664-235X
Online Access:Get full text
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Summary:In this paper, we for the first time get constructive solution for the inverse Sturm–Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction.
ISSN:1664-2368
1664-235X
DOI:10.1007/s13324-023-00845-3