# The exponent in the orthogonality catastrophe for Fermi gases

### Martin Gebert

King's College London, UK### Heinrich Küttler

Ludwig-Maximilians-Universität München, Germany### Peter Müller

Ludwig-Maximilians-Universität München, Germany### Peter Otte

FernUniversität Hagen, Germany

## Abstract

We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in $d$-dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting $N$-Fermion systems. We interpret the decay exponent $\gamma$ in terms of scattering theory and find $\gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\mathrm {HS}}^2$, where $T_E$ is the transition matrix at the Fermi energy $E$. This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352–359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.

## Cite this article

Martin Gebert, Heinrich Küttler, Peter Müller, Peter Otte, The exponent in the orthogonality catastrophe for Fermi gases. J. Spectr. Theory 6 (2016), no. 3, pp. 643–683

DOI 10.4171/JST/135