Description: Eigenfunctions of integrable planar billiards are studied — in particular,
the number of nodal domains, ν of the eigenfunctions
with Dirichlet boundary conditions are considered. The billiards
for which the time-independent Schrö dinger equation (Helmholtz
equation) is separable admit trivial expressions for the number
of domains. Here, we discover that for all separable and nonseparable
integrable billiards, ν satisfies certain difference equations.
This has been possible because the eigenfunctions can be
classified in families labelled by the same value ofm 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
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