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A nodal domain theorem for integrable billiards in two dimensions

Samajdar, Rhine and Jain, Sudhir R (2014) A nodal domain theorem for integrable billiards in two dimensions. In: ANNALS OF PHYSICS, 351 . pp. 1-12.

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Official URL: http://dx.doi.org/ 10.1016/j.aop.2014.08.010


Eigenfunctions of integrable planar billiards are studied - in particular, the number of nodal domains, nu of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrodinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and nonseparable integrable billiards, nu satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of m mod kn, given a particular k, for a set of quantum numbers, m, n. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations. (C) 2014 Elsevier Inc. All rights reserved.

Item Type: Journal Article
Additional Information: Copyright for this article belongs to the ACADEMIC PRESS INC ELSEVIER SCIENCE, 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
Keywords: Integrable billiards; Nodal domains; Quantum chaos
Department/Centre: Centres under the Director > Archives and Publication Cell
Date Deposited: 21 Jan 2015 06:30
Last Modified: 21 Jan 2015 06:30
URI: http://eprints.iisc.ac.in/id/eprint/50728

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