Inside-outside duality for planar billiards: A numerical study

Claude-Alain Pillet (1, 2) , Jean-Pierre Eckmann (3, 4) , Barbara Dietz (5) , Uzy Smilansky (6) , Iddo Ussishkin (6)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332
3 Département de Physique Théorique
4 Section de mathématiques [Genève]
5 LSP - Laboratoire de Spectrométrie Physique
6 Department of Physics of Complex Systems

This paper reports the results of extensive numerical studies related to spectral properties of the Laplacian and the scattering matrix for planar domains (called billiards). There is a close connection between eigenvalues of the billiard Laplacian and the scattering phases, which is, basically, that every energy at which a scattering phase is 2π corresponds to an eigenenergy of the Laplacian. Interesting phenomena appear when the shape of the domain does not allow an extension of the eigenfunction to the exterior. In this paper these phenomena are studied and illustrated from several points of view.

Mots clés: quantum billiards; inside-outside duality; scattering theory

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Claude-Alain Pillet, Jean-Pierre Eckmann, Barbara Dietz, Uzy Smilansky, Iddo Ussishkin. Inside-outside duality for planar billiards: A numerical study. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 1995, 51, pp.4222-4231. ⟨10.1103/PhysRevE.51.4222⟩. ⟨hal-00129121⟩