The boundary control approach to the Titchmarsh-Weyl $m-$function

TL;DR

This paper connects Boundary Control Theory with Titchmarsh-Weyl theory, offering a new perspective and efficient method for evaluating the m-function associated with Schrödinger operators on half-line domains.

Contribution

It introduces a novel interpretation of the A-amplitude and develops an efficient boundary control-based method for computing the Titchmarsh-Weyl m-function.

Findings

Provides a natural interpretation of the A-amplitude.

Develops an efficient computational method for the m-function.

Links boundary control and spectral theories for Schrödinger operators.

Abstract

We link the Boundary Control Theory and the Titchmarsh-Weyl Theory. This provides a natural interpretation of the $A-$amplitude due to Simon and yields a new efficient method to evaluate the Titchmarsh-Weyl $m-$function associated with the Schr\"{o}dinger operator $H=-\partial _{x}^{2}+q\left( x\right) $ on $L_{2}\left( 0,\infty \right) $ with Dirichlet boundary condition at $x=0.$