On the solution of the inverse problem for a class of canonical systems corresponding to matrix string equations

TL;DR

This paper addresses the inverse problem for a class of canonical systems linked to matrix string equations, providing solutions via spectral functions and $S$-nodes, and detailing procedures for inverse problem resolution.

Contribution

It introduces methods to solve inverse problems for canonical systems associated with matrix string equations using spectral functions and $S$-nodes, advancing theoretical understanding.

Findings

Explicit solutions for inverse problems are developed.

Procedures for reconstructing Hamiltonians from spectral data are provided.

Theoretical framework connects canonical systems with matrix string equations.

Abstract

We consider canonical systems (with $2p\times 2p$ Hamiltonians $H(x)\geq 0$), which correspond to matrix string equations. Direct and inverse problems are solved in terms of Titchmarsh--Weyl and spectral matrix functions and related $S$-nodes. Procedures for solving inverse problems are given.