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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| Дата: | 2012 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2012
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148652 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1486522019-02-19T01:24:52Z Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables Broadbridge, P. Chanu, C.M. Miller Jr., Willard 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. 2012 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/148652 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Broadbridge, P. Chanu, C.M. Miller Jr., Willard |
| spellingShingle |
Broadbridge, P. Chanu, C.M. Miller Jr., Willard Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Broadbridge, P. Chanu, C.M. Miller Jr., Willard |
| author_sort |
Broadbridge, P. |
| title |
Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables |
| 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 |
| publisher |
Інститут математики НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:30:58Z |
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