Quasi-exactly solvable potentials on the line and orthogonal polynomials
| dc.contributor.author | Finkel Morgenstern, Federico | |
| dc.contributor.author | González López, Artemio | |
| dc.contributor.author | Rodríguez González, Miguel Ángel | |
| dc.date.accessioned | 2023-06-20T20:08:47Z | |
| dc.date.available | 2023-06-20T20:08:47Z | |
| dc.date.issued | 1996-08 | |
| dc.description | ©1996 American Institute of Physics. IIt is a pleasure to thank M. A. Martín-Delgado, who pointed out to us reference [1], Gabriel Alvarez, for useful discussions regarding the theory of classical orthogonal polynomials, and Carlos Finkel, for providing several key references. The authors would also like to acknowledge the partial financial support of the GICYT under grant no. PB92-0197. | |
| dc.description.abstract | In this paper we show that a quasi-exactly solvable (normalizable or periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three-term recursion relation. in particular, we prove that (normalizable) exactly solvable one-dimensional systems are characterized by the fact that their associated polynomials satisfy a two-term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the kth moment grows Like the kth power of a constant as k tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems. | |
| dc.description.department | Depto. de Física Teórica | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | GICYT | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/31428 | |
| dc.identifier.doi | 10.1063/1.531591 | |
| dc.identifier.issn | 0022-2488 | |
| dc.identifier.officialurl | http://dx.doi.org/10.1063/1.531591 | |
| dc.identifier.relatedurl | http://scitation.aip.org | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/59672 | |
| dc.issue.number | 8 | |
| dc.journal.title | Journal of mathematical physics | |
| dc.language.iso | eng | |
| dc.page.final | 3972 | |
| dc.page.initial | 3954 | |
| dc.publisher | American Institute of Physics | |
| dc.relation.projectID | PB92-0197 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 51-73 | |
| dc.subject.ucm | Física-Modelos matemáticos | |
| dc.subject.ucm | Física matemática | |
| dc.title | Quasi-exactly solvable potentials on the line and orthogonal polynomials | |
| dc.type | journal article | |
| dc.volume.number | 37 | |
| dcterms.references | [1] Bender, C.M., and Dunne, G.V., Quasi-exactly solvable systems and orthogonal polynomials, J. Math. Phys. 37, 6 (1996). [2] Chihara, T.S., An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [3] Turbiner, A.V., Quasi-exactly solvable problems and sl(2) algebra, Commun. Math. Phys. 118, 467 (1988). [4] González-López, A., Kamran, N., and Olver, P.J., Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators, Commun. Math. Phys. 153, 117 (1993). [5] Krajewska, A., Ushveridze, A., and Walczak, Z., Bender-Dunne polynomials and quasi-exact solvabiltity, preprint, hep-th 9601088, 1996. [6] González-López, A., Kamran, N., and Olver, P.J., Quasi{exact solvability, Contemp. Math. 160, 113 (1994). [7] González-López, A., Kamran, N., and Olver, P.J., Quasi{exactly solvable Lie algebras of differential operators in two complex variables, J. Phys. A 24, 3995 (1991). [8] Kamran, N., and Olver, P.J., Lie algebras of differential operators and Lie-algebraic potentials, J. Math. Anal. Appl. 145, 342 (1990). [9] We assume that P is positive in some interval of the real line, so that the first integral in (12) is real. [10] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953. [11] Note, however, that a weakly orthogonal polynomial system need not satisfy a three-term recursion relation. [12] The moment problem in (-∞;∞) is never determined unless we impose the condition that ω is non-decreasing. Indeed, there are non-zero functions of bounded variation on (-∞;∞) whose moment functional (45) vanishes identically. [13] A classical theorem, [2], asserts that the moment problem of a positive-definite moment functional with bounded spectrum is determined; this theorem is not directly applicable in our case, however, since L is not positive-definite. [14] When the coefficient a_ k is positive for all k ≥ 1, several classical criteria due to Carleman relate the growth rate of the moments with the determinacy of the moment problem. Again, these results are not relevant here because of (43). [15] Peck, J.E.L., Polynomial curve ffitting with constraint, SIAM Review 4, 135 (1962). [16] Smirnov, Y., and Turbiner, A.V., Hidden sl(2) algebra of finite-difference equations, preprint, hep-th 9512002, 1995. [17] González-López, A., Kamran, N., and Olver, P.J., Real Lie algebras of differential operators and quasi-exactly solvable potentials, Phil. Trans. London Math. Soc. 354, 1165 (1996). | |
| dspace.entity.type | Publication | |
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| relation.isAuthorOfPublication.latestForDiscovery | 207092a4-0443-4336-a037-15936f8acc25 |
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