Direct spectral problems for Paley-Wiener canonical systems

TL;DR

This paper develops direct spectral problem techniques for Paley-Wiener (PW) canonical systems on $\mathbb{R}_+$, extending the existing PW-based inverse problem framework. By exploiting de Branges spaces and PW-measures, it establishes that a diagonal, step-function Hamiltonian with periodic PW-spectral data corresponds to orthogonal polynomials on the unit circle, enabling recovery via Verblunsky coefficients or spectral moments. It provides a finite-dimensional algorithm that reconstructs the spectral measure from the step-function data, and shows convergence of spectral measures for step-function approximations of non-step PW-Hamiltonians, including real Dirac systems. The results connect canonical systems to OPUC theory through explicit formulas and Toeplitz-determinant techniques, with concrete examples illustrating direct spectral construction and periodization limits. These findings advance the direct spectral theory for PW-class systems and enhance tractable approaches to spectral analysis in this setting.

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.

Figures (3)

• Figure 1: First $20$ terms of the cosine series of $\mu^T$ and $\mu$. Upper left: $T = \frac{1}{2}$, upper right: $T = \frac{1}{4}$, lower left: $T = \frac{1}{8}$, lower right: $T = \frac{1}{16}$.
• Figure 2: First $20$ terms of the cosine series of $\mu^T$. Blue: $T = \frac{1}{2}$. Orange: $T = \frac{1}{4}$. Yellow: $T = \frac{1}{8}$. Purple: $T = \frac{1}{16}$.
• Figure 3: $w^T(x)$ where $d^T\mu(x) = w^T(x) dx$. Blue: $T = \frac{1}{2}$. Orange: $T = \frac{1}{4}$. Purple: $T = \frac{1}{8}$. Black: $T = \frac{1}{16}$.

Theorems & Definitions (25)

• Theorem 2.1: dB
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6: MP
• Theorem 2.7: MP
• Remark 2.8
• Theorem 2.9
• Theorem 3.1: Verblunsky's Theorem