Quantum Ergodicity for Point Scatterers on Arithmetic Tori

TL;DR

This paper demonstrates quantum ergodicity for a Laplacian with a delta potential on the square torus, showing that a full density subsequence of new eigenfunctions equidistribute in phase space despite the absence of classical ergodicity.

Contribution

It establishes quantum ergodicity results for point scatterers on arithmetic tori, extending ergodic theory to non-ergodic classical systems with uniform estimates across coupling parameters.

Findings

Full density subsequence of new eigenfunctions equidistributes in phase space.

Equidistribution holds uniformly for both weak and strong coupling.

Results extend quantum ergodicity to non-ergodic classical systems.

Abstract

We prove an analogue of Shnirelman, Zelditch and Colin de Verdiere's Quantum Ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.