We give an upper bound on the modulus of the groundstate overlap of two non-interacting fermionic quantum systems with N particles in a large but finite volume L d of d-dimensional Euclidean space. The underlying one-particle Hamiltonians of the two systems are standard Schrödinger operators that differ by a non-negative compactly supported scalar potential. In the thermodynamic limit, the bound exhibits an asymptotic power-law decay in the system size L, showing that the ground-state overlap vanishes for macroscopic systems. The decay exponent can be interpreted in terms of the total scattering cross section averaged over all incident directions. The result confirms and generalises P. W. Anderson's informal computation [Phys. Rev. Lett. 18, 1049-1051.

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.

We derive rigorously the leading asymptotics of the so-called Anderson integral in the thermodynamic limit for one-dimensional, nonrelativistic, spin-less Fermi systems. The coefficient, γ, of the leading term is computed in terms of the S-matrix. This implies a lower and an upper bound on the exponent in Anderson's orthogonality catastrophe, CN −γ ≤ DN ≤ CN −γ pertaining to the overlap, DN , of ground states of non-interacting fermions.Mathematics Subject Classification (2010). Primary 81Q10, 34L40; Secondary 34L20, 34L25.