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.
Categories: math-ph math.MP math.SP