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

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 $\zeta (E):={\delta }^2(\sqrt{E})\mathrm{/}{\pi }^2$. 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.