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تسجيل مستخدم جديدSuperscars in the Seba billiard
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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.
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