English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We give a proof of the exact asymptotic behaviour in Anderson's OrthogonalityCatastrophe for Dirac-$\delta$ perturbations. We prove the exact asymptotics ofthe scalar product of the ground states of two non-interacting Fermi gasesconfined to a $3$-dimensional ball $B_L$ of radius $L$ in the thermodynamiclimit, where the underlying one-particle operators differ by a Dirac-$\delta$perturbation. More precisely, we show the algebraic decay of the correlationdeterminant $\big|\det\big(\langle\varphi_j^L,\psi_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-\zeta(E)+ \text{o}(1)}$, 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-particleSchr\"odinger operators. The decay exponent is given in terms of the s-wavescattering phase shift $\zeta(E):=\delta^2(\sqrt E)/{\pi^2}$. For an attractiveDirac-$\delta$ perturbation we conclude that the decay exponent $\frac 1{\pi^2}\Vert\arcsin |T(E)/2|\Vert^2_{\text{HS}}$ found in [GKMO14] does notprovide a sharp upper bound on the decay of the correlation determinant.

journal of mathematical physics

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