Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1996-03-15
J.Math.Phys. 37 (1996) 3954-3972
Physics
High Energy Physics
High Energy Physics - Theory
22 pages, plain TeX. Please typeset only the file orth.tex
Scientific paper
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 $k$-th moment grows like the $k$-th 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.
Finkel Federico
González-López Artemio
Rodriguez Miguel A.
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