An Abstract Approach to Weak Convergence of Spectral Shift Functions and Applications to Multi-Dimensional Schr\"odinger Operators

TL;DR

This paper develops an abstract framework demonstrating that spectral shift functions associated with sequences of self-adjoint operator pairs converge weakly, extending previous vague convergence results, with applications to multi-dimensional Schrödinger operators.

Contribution

It introduces a measure-theoretic approach to prove weak convergence of spectral shift functions, generalizing prior vague convergence results and applying to operators with various boundary conditions.

Findings

Spectral shift functions converge weakly in the abstract setting.

Vague convergence results extend to weak convergence.

Applications to multi-dimensional Schrödinger operators on expanding domains.

Abstract

We study the manner in which a sequence of spectral shift functions $\xi(\cdot;H_j,H_{0,j})$ associated with abstract pairs of self-adjoint operators $(H_j, H_{0,j})$ in Hilbert spaces $\cH_j$, $j\in\bbN$, converge to a limiting spectral shift function $\xi(\cdot;H,H_0)$ associated with a pair $(H,H_0)$ in the limiting Hilbert space $\cH$ as $j\to\infty$ (mimicking the infinite volume limit in concrete applications to multi-dimensional Schr\"odinger operators). Our techniques rely on a Fredholm determinant approach combined with certain measure theoretic facts. In particular, we show that prior vague convergence results for spectral shift functions in the literature actually extend to the notion of weak convergence. More precisely, in the concrete case of multi-dimensional Schr\"odinger operators on a sequence of domains $\Omega_j$ exhausting $\bbR^n$ as $j\to\infty$, we extend the convergence of associated spectral shift functions from vague to weak convergence and also from Dirichlet boundary conditions to more general self-adjoint boundary conditions on $\partial\Omega_j$.