RT Journal Article
T1 Quasi-exactly solvable potentials on the line and orthogonal polynomials
A1 Finkel Morgenstern, Federico
A1 González López, Artemio
A1 Rodríguez González, Miguel Ángel
AB 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.
PB American Institute of Physics
SN 0022-2488
YR 1996
FD 1996-08
LK https://hdl.handle.net/20.500.14352/59672
UL https://hdl.handle.net/20.500.14352/59672
LA eng
NO ©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.
NO GICYT
DS Docta Complutense
RD 9 abr 2025