Quasi-exactly solvable potentials on the line and orthogonal polynomials



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Finkel Morgenstern, Federico and González López, Artemio and Rodríguez González, Miguel Ángel (1996) Quasi-exactly solvable potentials on the line and orthogonal polynomials. Journal of mathematical physics, 37 (8). pp. 3954-3972. ISSN 0022-2488

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Official URL: http://dx.doi.org/10.1063/1.531591


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.

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©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.

Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:31428
Deposited On:15 Jul 2015 15:42
Last Modified:10 Dec 2018 15:10

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