Superscars in the Seba billiard
Henrik Ueberschaer, Par Kurlberg

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…
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Taxonomy
TopicsQuantum chaos and dynamical systems · Quantum Chromodynamics and Particle Interactions · Laser-Matter Interactions and Applications
