Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

Natalia P. Bondarenko and Andrey V. Gaidel
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Natalia P. Bondarenko: Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, 443086 Samara, Russia
Andrey V. Gaidel: Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, 410012 Saratov, Russia

Mathematics, 2021, vol. 9, issue 20, 1-25

Abstract: The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

Keywords: inverse spectral problem; quadratic differential pencil; global solvability; local solvability; stability; method of spectral mappings (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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