Solvability and stability of the inverse problem for the quadratic differential pencil

TL;DR

This paper investigates the inverse spectral problem for quadratic differential pencils, establishing conditions for solvability and stability, including complex coefficients and eigenvalue multiplicities, with a constructive approach and numerical validation.

Contribution

It provides new sufficient and necessary conditions for the global and local solvability of the inverse problem for quadratic differential pencils with complex coefficients.

Findings

Established conditions for global solvability

Proved local solvability and stability under perturbations

Developed a constructive method reducing the problem to a linear equation

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 nonlinear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.