Uniqueness theorems for meromorphic inner functions and canonical systems

TL;DR

The paper develops a comprehensive framework for uniqueness in the context of meromorphic inner functions (MIFs) on the upper half-plane by exploiting their Herglotz correspondence with meromorphic Herglotz functions and Clark measures. It establishes a suite of sharp uniqueness theorems that recover a MIF from spectral data, including spectra, derivative values at the spectrum, and Clark measure information, under natural summability or integrability conditions and with various normalization data. These results are then translated into inverse spectral statements for canonical Hamiltonian systems, yielding Borg-Levinson type theorems that recover the Hamiltonian from spectral measures or two spectra. The work bridges complex function theory and inverse spectral theory, providing robust criteria for unique Hamiltonian recovery and broad generalizations beyond classical Schrödinger operators.

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.

Theorems & Definitions (28)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Corollary 2.10