Uniform convergence of spectral shift functions

The spectral shift function \xi_{L}(E) for a Schr"odinger operator restricted to a finite cube of length L in multi-dimensional Euclidean space, with Dirichlet boundary conditions, counts the number of eigenvalues less than or equal to E \in \RR created by a perturbation potential V. We study the behavior of this function \xi_{L}(E) as L to infinity for the case of a compactly-supported and bounded potential V. After reviewing results of Kirsch [Proc. Amer. Math. Soc. 101, 509-512 (1987)], and our recent pointwise convergence result for the Ces`aro mean [Proc. Amer. Math. Soc. 138, 2141-2150 (2010)], we present a new result on the convergence of the energy-averaged spectral shift function that is uniform with respect to the location of the potential V within the finite box.