Etudes for the inverse spectral problem

TL;DR

This paper explores inverse spectral problems for canonical Hamiltonian systems, extending classical methods with new algorithms and examples, and incorporating truncated Toeplitz operators into the analysis.

Contribution

It broadens the scope of inverse spectral problem solutions for canonical systems by integrating truncated Toeplitz operators and providing new solution algorithms.

Findings

Extended classical inverse spectral methods to broader canonical systems.

Developed new algorithms for solving inverse spectral problems.

Illustrated solutions with diverse examples.

Abstract

In this note we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second order differential equations on a half-line. Our goal is to extend the classical resultss developed in the work of Marchenko, Gelfand-Levitan, and Krein to broader classes of canonical systems and to illustrate the solution algorithms and formulas with a variety of examples. One of the main ingredients of our approach is the use of truncated Toeplitz operators, which complement the standard toolbox of the Krein-de Branges theory of canonical systems.