Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2002-07-18
Nonlinear Sciences
Chaotic Dynamics
22 pages, 5 postscript figures. Part of fig. 1 is included in low resolution only. For a higher resolution version see http:
Scientific paper
10.1088/0305-4470/35/48/306
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 Poincar\'e 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.
Bäcker Arnd
Fürstberger S.
Schubert Roman
Steiner Frank
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