Behaviour of boundary functions for quantum billiards

Abstract

We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore they form a reduced representation of the quantum system, analogous to the Poincaré section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limacon billiards and the circle billiard, and good agreement is found. Furthermore we derive an asymptotic completeness relation for the boundary functions.

22 pages, 5 postscript figures. Part of fig. 1 is included in low resolution only. For a higher resolution version see http://www.physik.uni-ulm.de/theo/qc/

References in corpus (3)

• Autocorrelation function of eigenstates in chaotic and mixed systems
• Amplitude distribution of eigenfunctions in mixed systems
• About ergodicity in the family of limacon billiards

Cited by in corpus (6)

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• An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces
• Wavefunction Statistics using Scar States
• Reaction matrix for Dirichlet billiards with attached waveguides
• Completeness of boundary traces of eigenfunctions