The $m$-functions of discrete Schr\"odinger operators are sparse compared to those for Jacobi operators

TL;DR

This paper investigates the limited diversity of Weyl-Titchmarsh m-functions for discrete Schrödinger operators, highlighting their sparsity compared to Jacobi operators and implications for inverse spectral theory.

Contribution

It demonstrates the sparsity of m-functions for discrete Schrödinger operators using de Branges theory, explaining challenges in inverse spectral analysis.

Findings

m-functions of discrete Schrödinger operators are sparse

Their set cannot be dense in the set for Jacobi operators

Inverse spectral theory is more difficult due to this sparsity

Abstract

We explore the sparsity of Weyl-Titchmarsh $m$-functions of discrete Schr\"odinger operators. Due to this, the set of their $m$-functions cannot be dense on the set of those for Jacobi operators. All this reveals why an inverse spectral theory for discrete Schr\"odinger operators via their spectral measures should be difficult. To obtain the result, de Branges theory of canonical systems is applied to work on them, instead of Weyl-Titchmarsh $m$-functions.