Two inverse spectral problems for a class of singular Krein strings

TL;DR

This paper addresses inverse spectral problems for singular Krein strings, providing complete characterizations of spectral data and solutions via approximation methods, advancing understanding of spectral theory for these operators.

Contribution

It introduces solutions to inverse spectral problems for a class of singular Krein strings using approximation with Stieltjes strings, including a full description of spectral measures and spectra.

Findings

Complete characterization of spectral measures.

Solution to the inverse three-spectra problem.

Methodology based on approximation with Stieltjes strings.

Abstract

We solve the inverse problem from the spectral measure and the inverse three-spectra problem for the class of singular Krein strings on a finite interval with trace class resolvents. In particular, this includes a complete description of all possible spectral measures and three (Dirichlet) spectra associated with this class of Krein strings. The solutions of these inverse problems are obtained by approximation with Stieltjes strings.