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Superscars in the Šeba billiard
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 10, p. 2947-2964Article in journal (Refereed) Published
Abstract [en]

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 four wave vectors in momentum space-we therefore prove the existence of so-called "superscars" as predicted by Bogomolny and Schmit [5]. This result contrasts with the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus [14]. Further, in the strong coupling limit we show that a weaker form of localization holds for an essentially full density subsequence 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 andWinn who proved the existence of localized semiclassical measures under a clustering condition on the spectrum of the Laplacian.

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2017. Vol. 19, no 10, p. 2947-2964
Keywords [en]
Semiclassical measures, Superscars, Šeba billiard
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-217606DOI: 10.4171/JEMS/732ISI: 000415850700003Scopus ID: 2-s2.0-85032369129OAI: oai:DiVA.org:kth-217606DiVA, id: diva2:1157470
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 621-2011-5498
Note

QC 20171116

Available from: 2017-11-16 Created: 2017-11-16 Last updated: 2017-12-05Bibliographically approved

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