| 2. |
- Kurlberg, Pär, et al.
(författare)
-
Superscars in the Šeba billiard
- 2017
-
Ingår i: Journal of the European Mathematical Society (Print). - : European Mathematical Society Publishing House. - 1435-9855 .- 1435-9863. ; 19:10, s. 2947-2964
-
Tidskriftsartikel (refereegranskat)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 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.
|
|
