  We consider a multi-dimensional continuum Schr\"odinger operator $H$ which is
given by a perturbation of the negative Laplacian by a compactly supported
bounded potential. We show that, for a fairly large class of test functions,
the second-order Szeg\H{o}-type asymptotics for the spatially truncated Fermi
projection of $H$ is independent of the potential and, thus, identical to the
known asymptotics of the Laplacian.
