The asymptotics of an eigenfunction-correlation determinant for Dirac-$\delta$ perturbations (Anderson's Orthogonality Catastrophe for Dirac-$\delta$)

TL;DR

This paper rigorously determines the exact asymptotic decay rate of the eigenfunction-correlation determinant in Anderson's Orthogonality Catastrophe for Dirac-$oldsymbol{ extdelta}$ perturbations, revealing precise dependence on scattering phase shifts.

Contribution

It provides a rigorous proof of the exact asymptotics of the correlation determinant for Dirac-$oldsymbol{ extdelta}$ perturbations, refining previous bounds and clarifying the decay exponent in this context.

Findings

Exact algebraic decay rate of correlation determinant established.

Decay exponent expressed explicitly in terms of scattering phase shift.

Shows the previous upper bound is not sharp for attractive Dirac-$oldsymbol{ extdelta}$ perturbations.

Abstract

We give a proof of the exact asymptotic behaviour in Anderson's Orthogonality Catastrophe for Dirac-$\delta$ perturbations. We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a $3$-dimensional ball $B_L$ of radius $L$ in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-$\delta$ perturbation. More precisely, we show the algebraic decay of the correlation determinant $\big|\det\big(\langle\varphi_j^L, \psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}$, as $N,L\to\infty$ and $N/|B_L|\to~ \rho>0$, where $\varphi_j^L$ and $\psi_k^L$ denote the lowest-energy eigenfunctions of the finite-volume one-particle Schr\"odinger operators. The decay exponent is given in terms of the s-wave scattering phase shift $\zeta(E):=\delta^2(\sqrt E)/{\pi^2}$. For an attractive Dirac-$\delta$ perturbation we conclude that the decay exponent $\frac 1 {\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}$ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.