Superscars in the Seba billiard

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.

Quantum Ergodicity for Point Scatterers on Arithmetic Tori

We prove an analogue of Shnirelman, Zelditch and Colin de Verdieres 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.