On the weak and ergodic limit of the spectral shift function

TL;DR

This paper investigates how the spectral shift functions for certain Schrödinger operators behave as the interval length tends to infinity, focusing on their convergence properties in the weak and ergodic limits.

Contribution

It provides new insights into the convergence behavior of spectral shift functions for Schrödinger operators with Dirichlet boundary conditions as the domain expands.

Findings

Spectral shift functions converge in the weak limit as interval length increases.

Ergodic limit behavior of spectral shift functions is characterized.

Results contribute to understanding spectral properties of Schrödinger operators on unbounded domains.

Abstract

We discuss convergence properties of the spectral shift functions associated with a pair of Schrodinger operators with Dirichlet boundary conditions at the end points of a finite interval (0, r) as the length of interval approaches infinity.