The exponent in the orthogonality catastrophe for Fermi gases

Abstract

Dedicated to Hajo Leschke on the occasion of his 70 th birthday 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 finitevolume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding noninteracting N -particle systems. We interpret the decay exponent γ in terms of scattering theory and find γ = π −2 arcsin |T E /2| 2 HS , 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.

Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function

“…Let (L n ) n∈N be a sequence of lengths such that L n / e n α → ∞ as n → ∞ for some α > 1. Then, Theorem 2.2 in [GKMO16], see also [GKM14], applies realisationwise to the operators H and H τ for any τ ∈ (0, 1], and we infer that almost surely and for a.e. energy E ∈ J S Ln (E) L −γ(E)/2+o(L 0 n ) n as n → ∞.…”

“…for every n ∈ N. Now, suppose we knew there exists a spectral interval J ⊂ R such that H has absolutely continuous spectrum in J almost surely. It follows from (3.11) and [GKMO16,Thm. 3.4] that, almost surely given any p > 0, 1 (−∞,E] (H L ) − 1 (−∞,E] (H τ L ) p diverges at least logarithmically in L for a.e.…”

The spectral density of a product of spectral projections

“…This is essentially a known statement (see e.g. [Pu,Lemma 4] or [GKMO,Corollary 4.31]). For completeness, we briefly recall the proof.…”

On an Integral Formula for Fredholm Determinants Related to Pairs of Spectral Projections

“…holds and therefore multiplying the latter with its adjoint we obtain . In this situation, the above determinant (2.11) is related to the scalar product of the ground states of two non-interacting fermionic systems at Fermi energy E, see [9] and references cited therein. The above formula can be used to compute this scalar product exactly in cases where the Fermi energy lies in a spectral gap.…”

“…Over the last years further interest in Fredholm determinants of the form (1.1) emerged in mathematical physics, see [13,8,9,6,14,4]. The determinant (1.1) appears when computing the thermodynamic limit of the scalar product of two non-interacting fermionic many-body ground states filled up to the Fermi energy E ∈ R. In this case, the underlying one-particle operators are given by a pair of Schrödinger operators whose difference is relatively trace class, i.e.…”