Solutions of Helmholtz and Schrödinger equations with side condition and nonregular separation of variables

Philip Broadbridge, Claudia M. Chanu, Willard Miller

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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 Ndimensional 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.

Original languageEnglish (US)
Article number089
JournalSymmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume8
DOIs
StatePublished - 2012

Keywords

  • Helmholtz equation
  • Nonregular separation of variables
  • Schrödinger equation

Fingerprint Dive into the research topics of 'Solutions of Helmholtz and Schrödinger equations with side condition and nonregular separation of variables'. Together they form a unique fingerprint.

Cite this