Periodic approximations in inverse spectral problems for canonical Hamiltonian systems

TL;DR

This paper extends a periodic approximation method for inverse spectral problems of canonical Hamiltonian systems to non-periodic measures by using periodizations and convergence analysis.

Contribution

It introduces a novel approach to approximate Hamiltonians for non-periodic spectral measures through periodizations and convergence techniques.

Findings

The periodic algorithm is extended to non-periodic measures.

Hamiltonians from periodizations converge to the original Hamiltonian.

The method broadens the applicability of inverse spectral problem solutions.

Abstract

This note is devoted to inverse spectral problems for canonical Hamiltonian systems on the half-line. An approach to inverse spectral problems based on the use of truncated Toeplitz operators has been especially effective in the case when the spectral measure of the system is a locally finite periodic measure (see \cite{MP}). In this note we extend the periodic algorithm to the case of non-periodic measures by considering periodizations of a spectral measure and showing that the Hamiltonians corresponding to the periodizations converge to the Hamiltonian of the original measure.