The spectral shift function for compactly supported perturbations of Schrödinger operators on large bounded domains

We study the asymptotic behavior as $L \rightarrow \infty$ of the finite-volume spectral shift function for a positive, compactly supported perturbation of a Schrödinger 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 Cesàro mean of finite-volume spectral shift functions remains pointwise bounded along certain sequences $L_n \rightarrow \infty$ 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 $L \rightarrow \infty$. 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.