### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 089, 31 pages      arXiv:1209.2019      https://doi.org/10.3842/SIGMA.2012.089
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

### Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables

Philip Broadbridge a, Claudia M. Chanu b and Willard Miller Jr. c
a) School of Engineering and Mathematical Sciences, La Trobe University, Melbourne, Australia
b) Dipartimento di Matematica G. Peano, Università di Torino, Torino, Italy
c) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Received September 21, 2012, in final form November 19, 2012; Published online November 26, 2012

Abstract
Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schrödinger equations with potential on N-dimensional Riemannian and pseudo-Riemannian manifolds, but with a linear side condition, where more structure is available. We show that the requirement of N−1 commuting second-order symmetry operators, modulo a second-order linear side condition corresponds to nonregular separation of variables in an orthogonal coordinate system, characterized by a generalized Stäckel matrix. The coordinates and solutions obtainable through true nonregular separation are distinct from those arising through regular separation of variables. We develop the theory for these systems and provide examples.

Key words: nonregular separation of variables; Helmholtz equation; Schrödinger equation.

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