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 SIGMA, 2012, Volume 8, 089, 31 pages (Mi sigma766)

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

Philip Broadbridgea, Claudia M. Chanub, 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

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.

Keywords: nonregular separation of variables; Helmholtz equation; Schrödinger equation

DOI: https://doi.org/10.3842/SIGMA.2012.089

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ArXiv: 1209.2019
MSC: 35Q40; 35J05
Received: September 21, 2012; in final form November 19, 2012; Published online November 26, 2012
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Citation: Philip Broadbridge, Claudia M. Chanu, Willard Miller Jr., “Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables”, SIGMA, 8 (2012), 089, 31 pp.

Citation in format AMSBIB
\Bibitem{BroChaMil12} \by Philip~Broadbridge, Claudia~M.~Chanu, Willard~Miller~Jr. \paper Solutions of Helmholtz and Schr\"odinger Equations with Side Condition and Nonregular Separation of Variables \jour SIGMA \yr 2012 \vol 8 \papernumber 089 \totalpages 31 \mathnet{http://mi.mathnet.ru/sigma766} \crossref{https://doi.org/10.3842/SIGMA.2012.089} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3007270} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000312378600001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84870309828} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Andrey V. Tsiganov, “On a Trivial Family of Noncommutative Integrable Systems”, SIGMA, 9 (2013), 015, 13 pp.
2. Miller Jr. Willard, Turbiner A.V., “Particle in a Field of Two Centers in Prolate Spheroidal Coordinates: Integrability and Solvability”, J. Phys. A-Math. Theor., 47:19 (2014), 192002
3. Gaeta G., “Symmetry and Lie–Frobenius Reduction of Differential Equations”, J. Phys. A-Math. Theor., 48:1 (2015), 015202
4. Kholodenko A.L., Kauffman L.H., “Huygens Triviality of the Time-Independent Schrodinger Equation. Applications to Atomic and High Energy Physics”, Ann. Phys., 390 (2018), 1–59
5. Claudia Maria Chanu, Giovanni Rastelli, “Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians”, SIGMA, 15 (2019), 013, 22 pp.
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