Statistics of Wave Functions for a Point Scatterer on the Torus

Zeév Rudnick, Henrik Ueberschär

Research output: Contribution to journalArticlepeer-review

Abstract

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.

Original languageEnglish
Pages (from-to)763-782
Number of pages20
JournalCommunications in Mathematical Physics
Volume316
Issue number3
DOIs
StatePublished - Dec 2012

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