Publication details
- Title:
- Behaviour of boundary functions for quantum billiards
- Author:
- A. Bäcker, S. Fürstberger, R. Schubert and F. Steiner
- Journal:
-
J. Phys. A 35 (2002) 10293-10310
- 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 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 limaçon
billiards and the circle billiard, and good agreement is
found.
Furthermore we derive an asymptotic completeness relation for
the boundary functions.
- Download:
- Please refer to the above published version.
(The preprint version of this article can be obtained as
Ulm report ULM-TP/02-6
(July 2002), 21 pp. (316KB compressed, 981KB uncompressed).)
Last modified: 27 October 2004, 16:21:49
Impressum, © Arnd Bäcker
