The spectral shift function for compactly supported perturbations of Schr\"odinger operators on large bounded domains

TL;DR

This paper investigates the asymptotic behavior of the spectral shift function for large domains with compactly supported perturbations of Schrödinger operators, proving convergence properties and providing new proofs of existing formulas.

Contribution

It establishes the pointwise boundedness of the Cesàro mean of spectral shift functions and offers a new proof of the Birman–Solomyak formula for these functions.

Findings

Cesàro mean of spectral shift functions remains bounded for almost every energy.

Vague convergence of finite-volume to infinite-volume spectral shift functions is proven.

New proof of the Birman–Solomyak formula relating spectral shift functions to the potential.

Abstract

We study the asymptotic behavior as L \to \infty of the finite-volume spectral shift function for a positive, compactly-supported perturbation of a Schr\"odinger operator in d-dimensional Euclidean space, restricted to a cube of side length L with Dirichlet boundary conditions. The size of the support of the perturbation is fixed and independent of L. We prove that the Ces\`aro mean of finite-volume spectral shift functions remains pointwise bounded along certain sequences L_n \to \infty for Lebesgue-almost every energy. In deriving this result, we give a short proof of the vague convergence of the finite-volume spectral shift functions to the infinite-volume spectral shift function as L \to\infty . Our findings complement earlier results of W. Kirsch [Proc. Amer. Math. Soc. 101, 509 - 512 (1987), Int. Eqns. Op. Th. 12, 383 - 391 (1989)] who gave examples of positive, compactly-supported perturbations of finite-volume Dirichlet Laplacians for which the pointwise limit of the spectral shift function does not exist for any given positive energy. Our methods also provide a new proof of the Birman--Solomyak formula for the spectral shift function that may be used to express the measure given by the infinite-volume spectral shift function directly in terms of the potential.