An inverse problem for a class of canonical systems and its applications to self-reciprocal polynomials

TL;DR

This paper develops an explicit method to recover Hamiltonians of canonical systems from Hermite-Biehler class functions and applies it to analyze the roots of self-reciprocal polynomials.

Contribution

It provides a new explicit approach to reconstruct Hamiltonians from exponential polynomials in the Hermite-Biehler class, advancing inverse spectral theory.

Findings

Explicit Hamiltonian recovery method from exponential polynomials

Application to root distribution of self-reciprocal polynomials

Enhanced understanding of canonical systems and their inverse problems

Abstract

A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This type inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials.