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

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 $\left|\det \right(⟨{\phi }_j^L,{\psi }_k^L⟩{)}_{j,k=1,...,N}{|}^2=L^{-\zeta (E)+\text{o}(1)}$, as $N,L\to \infty$ and $N/|B_L|\to \rho >0$, where ${\phi }_j^L$ and ${\psi }_k^L$ 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 $\zeta (E):={\delta }^2(\sqrt{E})/{\pi }^2$. For an attractive Dirac-$\delta$ perturbation we conclude that the decay exponent $\frac{1}{{\pi }^2}\Vert \arcsin |T(E)/2|{\Vert }_{\text{HS}}^2$ found in [GKMO14] does not provide a sharp upper bound on the decay of the correlation determinant.

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References in corpus (3)

• Anderson's orthogonality catastrophe
• Anderson's orthogonality catastrophe in one dimension induced by a magnetic field
• Finite-size energy of non-interacting Fermi gases

Cited by in corpus (3)

• Finite-size energy of non-interacting Fermi gases
• Ground-state-energy universality of noninteracting fermionic systems
• On an integral formula for Fredholm determinants related to pairs of spectral projections