Uniform convergence of spectral shift functions

The spectral shift function ξL(E) for a Schrödinger 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 ∈ R created by a perturbation potential V . We study the behavior of this function ξL(E) as L→∞ 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 Cesàro 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. § 1. Statement of the Problem and Result In the late eighties, W. Kirsch [11, 12] investigated the relative eigenvalue counting function ξL(E) for compactly-supported nonnegative perturbations V of the nonnegative Laplacian −∆L > 0 on cubes ΛL :=] − L/2, L/2[⊂ R with Dirichlet boundary conditions. The cubes ΛL of edge length L > 0 are centered about the origin in ddimensional Euclidean space R. The real-valued local perturbation V is a compactlysupported, nonnegative potential 0 6 V ∈ L∞(Rd). We emphasize that V does not depend on L and, of course, L is large enough so that supp(V ) ⊂ ΛL. The finite-volume spectral shift function (SSF) or relative eigenvalue counting function ξL(E) for the pair of self-adjoint operators (−∆L/2 + V,−∆L/2) on the Hilbert Version of July 15, 201