Superscars in the Seba billiard

TL;DR

This paper investigates eigenfunctions of the Laplacian with a point scatterer on an irrational torus, revealing the existence of superscars and localization phenomena in certain eigenfunctions, contrasting with phase space equidistribution.

Contribution

It characterizes semiclassical measures of eigenfunctions affected by a point scatterer on an irrational torus, demonstrating superscars and localization in the weak and strong coupling regimes.

Findings

Existence of superscars localized along 4 wave vectors in momentum space.

Full density subset of eigenfunctions exhibits phase space equidistribution.

Positive proportion of eigenvalues show localization, violating quantum ergodicity.

Abstract

We consider the Laplacian with a delta potential (a "point scatterer") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes ---"old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along 4 wave vectors in momentum space --- we therefore prove the existence of so-called "superscars" as predicted by Bogomolny and Schmit. This result contrasts the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus. Further, in the strong coupling limit we show that a weaker form of localization holds for a positive proportion of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof and Winn who proved the existence of localized semiclassical measures under a non-clustering condition on the spectrum of the Laplacian.