TY - JOUR
T1 - Solutions of Helmholtz and Schrödinger equations with side condition and nonregular separation of variables
AU - Broadbridge, Philip
AU - Chanu, Claudia M.
AU - Miller, Willard
PY - 2012
Y1 - 2012
N2 - 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.
AB - 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.
KW - Helmholtz equation
KW - Nonregular separation of variables
KW - Schrödinger equation
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U2 - 10.3842/SIGMA.2012.089
DO - 10.3842/SIGMA.2012.089
M3 - Article
AN - SCOPUS:84870309828
SN - 1815-0659
VL - 8
JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
M1 - 089
ER -