Half line Titchmarsh-Weyl $m$ functions of vector-valued discrete Schrodinger operators

TL;DR

This paper studies the properties of half-line Titchmarsh-Weyl $m$ functions for vector-valued discrete Schrödinger operators, showing they lie in the Siegel upper half space and analyzing their behavior under transfer matrices.

Contribution

It introduces a metric on the space of $m$ functions and proves that transfer matrix actions are distance decreasing in this metric.

Findings

$m$ functions are elements of the Siegel upper half space.

Transfer matrices act as distance decreasing transformations.

The metric provides a new way to analyze spectral properties.

Abstract

We show that the half-line $m$ functions associated with the vector-valued Schrodinger operators are the elements in the Siegel upper half space. We introduce a metric on the space of $m$ functions associated to the vector-valued discrete Schrodinger operators. Then we show that the action of transfer matrices on these $m$ functions is distance decreasing.