Uniqueness theorems for meromorphic inner functions and canonical systems

TL;DR

This paper establishes uniqueness theorems for meromorphic inner functions and applies these results to inverse spectral problems of canonical Hamiltonian systems, including generalizations of classical spectral theorems.

Contribution

It introduces new uniqueness results for meromorphic inner functions and extends inverse spectral theory for canonical Hamiltonian systems.

Findings

Proved uniqueness theorems for meromorphic inner functions based on spectral data.

Generalized Borg-Levinson two-spectra theorem for canonical Hamiltonian systems.

Achieved unique determination of Hamiltonians from spectral measures under specific conditions.

Abstract

We prove uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems we consider spectral data depending partially or fully on the spectrum, derivative values at the spectrum, Clark measure or the spectrum of the negative of a meromorphic inner function. Moreover we consider applications of these uniqueness results to inverse spectral theory of canonical Hamiltonian systems and obtain generalizations of Borg-Levinson two-spectra theorem for canonical Hamiltonian systems and unique determination of a Hamiltonian from its spectral measure under some conditions.