Direct spectral problems for Paley-Wiener canonical systems

TL;DR

This paper develops a method to solve the direct spectral problem for Paley-Wiener canonical systems by approximating general Hamiltonians with step-functions and analyzing the convergence of their spectral measures.

Contribution

It introduces a new approach to the direct spectral problem for canonical systems using step-function approximations and spectral measure convergence analysis.

Findings

Spectral measures of step-function Hamiltonians converge to that of the original Hamiltonian.

Reversal of the inverse spectral problem algorithm for step-function Hamiltonians.

Demonstrates the effectiveness of spectral measure approximation for non-step Hamiltonians.

Abstract

This note focuses on the direct spectral problem for canonical Hamiltonian systems on the half-line $\mathbb{R}_+$. Truncated Toeplitz operators have been effectively used to solve the inverse spectral problem when the spectral measure is a locally finite periodic measure (see \cite{MP}). Here, we reverse the inverse problem algorithm to solve the direct spectral problem for step-function Hamiltonians. For a non-step-function Hamiltonian, we consider its step-function approximations and their corresponding spectral measures, and show that these spectral measures converge to the spectral measure of the original Hamiltonian.