Problems in spectral analysis of canonical Hamiltonian systems

TL;DR

This paper reviews recent spectral analysis results for canonical Hamiltonian systems, highlighting inverse spectral problems, connections with classical analysis tools, and applications to Schrödinger operators.

Contribution

It advances understanding of inverse spectral problems for canonical systems and links them with classical analysis methods, providing new insights and examples.

Findings

Solutions to inverse spectral problems for canonical systems

Connections between spectral analysis and classical analysis tools

Illustrative examples demonstrating theoretical results

Abstract

This note focuses on recent results in spectral analysis of canonical systems of differential equations obtained via the approach developed in our previous papers \cite{MIF1, MP3, etudes, etudes2, PZ, Direct}. Many of our results are motivated by the pioneering research of Barry Simon and his co-authors; see, for instance, the papers cited in the main text. We discuss solutions to the inverse spectral problem (ISP) for canonical Hamiltonian systems and mixed spectral problems for Schr\"odinger operators. One of our goals is to show connections of ISP with classical tools of analysis, such as the Hilbert transform, orthogonal polynomials, the gap problem and solutions to the Riemann-Hilbert problem. We illustrate our results with examples and discuss further questions.