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Behaviour of boundary functions for quantum billiards

A Bäcker1, S Fürstberger1, R Schubert2 and F Steiner1

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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 limaçon billiards and the circle billiard, and good agreement is found. Furthermore, we derive an asymptotic completeness relation for the boundary functions.


PACS

03.65.Sq Semiclassical theories and applications

02.10.Ud Linear algebra

03.65.Fd Algebraic methods

02.30.Rz Integral equations

05.45.Mt Quantum chaos; semiclassical methods

02.30.Jr Partial differential equations

MSC

82C05 Classical dynamic and nonequilibrium statistical mechanics (general)

81Q20 Semiclassical techniques including WKB and Maslov methods

45C05 Eigenvalue problems (See also 34Lxx, 35Pxx, 45P05, 47A75)

82-08 Computational methods

81Q50 Quantum chaos (See also 37Dxx)

65R20 Integral equations

Subjects

Mathematical physics

Statistical physics and nonlinear systems

Quantum information and quantum mechanics

Dates

Issue 48 (6 December 2002)

Received 19 July 2002

Published 19 November 2002



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