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

We study the asymptotic behavior as L→∞ of the finite-volume spectral shift function for a positive, compactly supported perturbation of a Schrodinger 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 Cesaro mean of finite-volume spectral shift functions remains pointwise bounded along certain sequences L_n → ∞ 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 → ∞. 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.