Stability of a Szeg\H{o}-type asymptotics

TL;DR

This paper proves that for a class of Schrödinger operators with compactly supported potentials, the second-order Szegő-type asymptotics of the Fermi projection are identical to those of the free Laplacian, indicating potential independence.

Contribution

It establishes potential independence of second-order Szegő asymptotics for a class of multi-dimensional Schrödinger operators.

Findings

Asymptotics are identical to the Laplacian case.

Second-order Szegő asymptotics are independent of the potential.

Results apply to a broad class of test functions.

Abstract

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.