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

We give a proof of the exact asymptotic behaviour in Anderson's Orthogonality Catastrophe for Dirac-$δ$ 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-$δ$ perturbation. More precisely, we show the algebraic decay of the correlation determinant $\big|\det\big(\langleφ_j^L, ψ_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-ζ(E)+ \text{o}(1)}$, as $N,L\to\infty$ and $N/|B_L|\to~ ρ>0$, where $φ_j^L$ and $ψ_k^L$ denote the lowest-energy eigenfunctions of the finite-volume one-particle Schrödinger operators. The decay exponent is given in terms of the s-wave scattering phase shift $ζ(E):=δ^2(\sqrt E)/{π^2}$. For an attractive Dirac-$δ$ perturbation we conclude that the decay exponent $\frac 1 {π^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}$ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.

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