The One-Dimenshional Inverse Wave Spectral Problem with Discontinuous Wave Speed

TL;DR

This paper investigates the inverse spectral problem for a Sturm-Liouville operator with discontinuous wave speed, establishing uniqueness and providing a constructive solution method for complex periodic potentials.

Contribution

It introduces a new approach to solve the inverse problem with discontinuous coefficients, including a uniqueness theorem and a constructive solution procedure.

Findings

Spectrum characterization for operators with discontinuous coefficients

Proof of uniqueness for the inverse problem

Constructive method for reconstructing the potential

Abstract

The inverse problem for the Sturm- Liouville operator with complex periodic potential and positive 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