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Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces

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dc.contributor.authorGómez-Ullate Otaiza, David
dc.contributor.authorKamran, Niky
dc.contributor.authorMilson, Robert
dc.date.accessioned2023-06-20T10:55:12Z
dc.date.available2023-06-20T10:55:12Z
dc.date.issued2007-05
dc.description© American Institute of Mathematical Sciences.
dc.description.abstractIn this paper we derive structure theorems which characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write an explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, especially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.
dc.description.departmentDepto. de Física Teórica
dc.description.facultyFac. de Ciencias Físicas
dc.description.refereedTRUE
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/30879
dc.identifier.doi10.3934/dcds.2007.18.85
dc.identifier.issn1078-0947
dc.identifier.officialurlhttp://dx.doi.org/10.3934/dcds.2007.18.85
dc.identifier.relatedurlhttp://www.aimsciences.org
dc.identifier.relatedurlhttp://arxiv.org/abs/nlin/0604070
dc.identifier.urihttps://hdl.handle.net/20.500.14352/51451
dc.issue.number1
dc.journal.titleDiscrete and continuous dynamical systems
dc.language.isoeng
dc.page.final106
dc.page.initial85
dc.publisherAmerican Institute of Mathematical Sciences
dc.rights.accessRightsopen access
dc.subject.cdu51-73
dc.subject.keywordDiffusion-equations
dc.subject.keywordCalogero
dc.subject.keywordAlgebras
dc.subject.keywordSpaces
dc.subject.ucmFísica-Modelos matemáticos
dc.subject.ucmFísica matemática
dc.titleStructure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces
dc.typejournal article
dc.volume.number18
dcterms.references[1] M. Shifman and A. Turbiner, Quantal problems with partial algebraization of the spectrum, Commun. Math. Phys. 126 (1989) 347-365. [2] A. Turbiner, Quasi-exactly-solvable problems and sl(2) algebra, Comm. Math. Phys. 118 (1988), no. 3, pp. 467–474. [3] A. Ushveridze, “Quasi-exactly-solvable models in quantum mechanics”, Institute of Physics Publishing: Bristol 1994. [4] N. Kamran and P.J. Olver, Lie algebras of differential operators and Lie-algebraic potentials, J. Math. Anal. Appl. 145 (1990), no. 2, 342–356. [5] F. Calogero, “Classical many-body problems amenable to exact treatments”, Lecture Notes in Physics Monograph m 66, Springer, Berlin, 2001. [6] G. Post and A. Turbiner, Classification of linear differential operators with an invariant subspace of monomials, Russian J. Math. Phys. 3 (1995), no. 1, 113–122. [7] D. Gómez-Ullate, N. Kamran, and R. Milson, Quasi-exact solvability and the direct approach to invariant subspaces, J. Phys. A 38 (2005), 2005–2019. [8] D. Gómez-Ullate, N. Kamran, and R. Milson, Quasi-exact solvability beyond the SL(2) algebraization, arXiv:nlin.SI/0601053. [9] F. Finkel and N. Kamran, The Lie algebraic structure of differential operators admitting invariant spaces of polynomials, Adv. in Appl. Math. 20 (1998), no. 3, 300–322. [10] R. Milson, On the construction of quasi-exactly solvable Schrodinger operators on homogeneous spaces, J. Math. Phys. 36 (1995), 6004–6027. [11] L. Brink, A. Turbiner and N. Wyllard, Hidden algebras of the (super) Calogero and Sutherland models, J. Math. Phys. 39 (1998), 1285–1315. [12] D. Gómez-Ullate, A. Gónzalez-López and M.A. Rodríguez, Exact solutions of a new elliptic Calogero–Sutherland model, Phys. Lett. B511 (2001), 112–118. [13] F. Finkel, D. Gómez-Ullate, A. González-López, M.A. Rodríguez and R, Zhdanov, AN – type Dunkl operators and new spin Calogero–Sutherland models, Commun. Math. Phys. 221 (2001), 477–497. [14] N. Kamran, R. Milson and P.J. Olver, Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, Adv. Math. 156 (2000), no. 2, 286–319. [15] V.A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 2, 225–246. [16] J.R. King, Exact multidimensional solutions to some nonlinear diffusion equations, Quart. J. Mech. Appl. Math. 46 (1993), no. 3, 419–436. [17] J.R. King, Two generalisations of the thin film equation, Math. Comput. Model. 34 (2001), 737–756. [18] J.R. King, Exact polynomial solutions to some nonlinear diffusion equations, Physica D 64 (1993), 35–65. [19] S. Svirshchevskii, Ordinary differential operators possessing invariant subspaces of polynomial type, Commun. Nonlinear Sci. Numer. Simul. 9 (2004), no. 1, 105–115.
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