Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables
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 Schröd...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2012 |
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| Format: | Artikel |
| Sprache: | English |
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Інститут математики НАН України
2012
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148652 |
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| Zitieren: | Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables / P. Broadbridge, C.M. Chanu, Willard Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 36 назв. — англ. |
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Broadbridge, P. Chanu, C.M. Miller Jr., Willard 2019-02-18T17:33:50Z 2019-02-18T17:33:50Z 2012 Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables / P. Broadbridge, C.M. Chanu, Willard Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 36 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35Q40; 35J05 DOI: http://dx.doi.org/10.3842/SIGMA.2012.089 https://nasplib.isofts.kiev.ua/handle/123456789/148652 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. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html. This work was partially supported by a grant from the Simons Foundation (#208754 to Willard Miller, Jr.) and by the Australian Research Council (grant DP1095044 to G.E. Prince and P. Broadbridge). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables |
| spellingShingle |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables Broadbridge, P. Chanu, C.M. Miller Jr., Willard |
| title_short |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables |
| title_full |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables |
| title_fullStr |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables |
| title_full_unstemmed |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables |
| title_sort |
solutions of helmholtz and schrödinger equations with side condition and nonregular separation of variables |
| author |
Broadbridge, P. Chanu, C.M. Miller Jr., Willard |
| author_facet |
Broadbridge, P. Chanu, C.M. Miller Jr., Willard |
| publishDate |
2012 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148652 |
| citation_txt |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables / P. Broadbridge, C.M. Chanu, Willard Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 36 назв. — англ. |
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2025-12-07T16:00:09Z |
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