Density of Schr\"odinger Weyl-Titchmarsh m functions on Herglotz functions

TL;DR

This paper proves that Schrödinger Weyl-Titchmarsh m functions are densely distributed among all Herglotz functions, using de Branges theory, enhancing understanding of spectral properties of one-dimensional Schrödinger operators.

Contribution

It demonstrates the density of Schrödinger Weyl-Titchmarsh m functions within the space of all Herglotz functions, connecting spectral theory with de Branges' canonical systems.

Findings

Schrödinger Weyl-Titchmarsh m functions are dense in Herglotz functions.

The result is achieved via de Branges theory.

The density holds with respect to uniform convergence on compact sets.

Abstract

We show that the Herglotz functions that arise as Weyl-Titchmarsh $m$ functions of one-dimensional Schr\"odinger operators are dense in the space of all Herglotz functions with respect to uniform convergence on compact subsets of the upper half plane. This result is obtained as an application of de Branges theory of canonical systems.