  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.
