A difference-equation formalism for the nodal domains of separable billiards

Abstract

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

Raising and lowering operators for quantum billiards

“…Within each class, it was shown that the number of domains, ν m,n for one eigenfunction is related to ν m+2n,n by a difference equation [9,7]. We can, in fact, write down the operator (in the matrix form) which actually takes us along the ladder of states beginning with ψ m,n , up and down.…”

“…There are some very interesting connections between exactly solvable models and random matrix theories, a summary may be seen in [6]. The Helmholtz operator is separable in certain coordinate systemsfor these cases, the solutions can be found [7]. The non-separable problems for which the classical dynamics is integrable have been recently studied in detail [8,9,10].…”

“…For instance, the solutions of the Helmholtz equation for a circular, elliptical, circular annulus, elliptical annulus, confocal parabolic enclosures are each a product of functions like Bessel for circular, Mathieu for elliptic and so on [7].…”

“…Even if we restrict to systems that are classically integrable, the problem poses considerable challenge. Progress on this otherwise intractable problem could be made recently due to the observation that the eigenfunctions could be classified in terms of equivalence classes [9,7]. Fig.…”

“…Although the solutions of these systems have been known, there remain many questions regarding the nature of nodal curves and domains. The nodal domains of the eigenfunctions of the systems for which the Schrödinger equation is separable, form a checkerboard pattern [7]. the number of crossings actually count the number of domains.…”