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Journal of Spectral Theory

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Volume 6, Issue 3, 2016, pp. 643–683
DOI: 10.4171/JST/135

Published online: 2016-09-15

The exponent in the orthogonality catastrophe for Fermi gases

Martin Gebert[1], Heinrich Küttler[2], Peter Müller[3] and Peter Otte[4]

(1) King's College London, UK
(2) Ludwig-Maximilians-Universität München, Germany
(3) Ludwig-Maximilians-Universität München, Germany
(4) FernUniversität Hagen, Germany

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.

Keywords: Schrödinger operators, Anderson orthogonality, spectral correlations, scattering theory

Gebert Martin, Küttler Heinrich, Müller Peter, Otte Peter: The exponent in the orthogonality catastrophe for Fermi gases. J. Spectr. Theory 6 (2016), 643-683. doi: 10.4171/JST/135