Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil

TL;DR

This paper investigates the spectral properties and inverse problem solutions for a class of second-order indefinite non-self-adjoint differential operator pencils with complex periodic potentials and discontinuous coefficients, providing theoretical foundations and constructive methods.

Contribution

It introduces a new inverse problem formulation, proves uniqueness, and offers a constructive solution method for a specific class of complex differential operator pencils.

Findings

Spectrum characterized for the operator pencil

Uniqueness theorem established for the inverse problem

Constructive procedure for solving the inverse problem provided

Abstract

The inverse problem for the differential operator pencil with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem