Weak Convergence of Spectral Shift Functions for One-Dimensional Schr\"odinger Operators

TL;DR

This paper investigates how spectral shift functions for one-dimensional Schrödinger operators on finite intervals converge to the half-line spectral shift function as the interval length tends to infinity, extending previous vague convergence results.

Contribution

It extends existing vague convergence results to weak convergence for spectral shift functions under various boundary conditions using a Fredholm determinant approach.

Findings

Spectral shift functions converge weakly in the infinite volume limit.

Results apply to arbitrary separated self-adjoint boundary conditions.

The approach generalizes previous convergence results for Dirichlet conditions.

Abstract

We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schr\"odinger operators on the finite interval $(0,R)$ converge in the infinite volume limit $R\to\infty$ to the half-line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at $x=0$ and $x=R$.