QUASI-EXACTLY SOLVABLE POTENTIALS ON THE LINE AND ORTHOGONAL POLYNOMIALS

In this paper we show that a quasi-exactly solvable (normalizable or p eriodic) one-dimensional Hamiltonian satisfying very mild conditions d efines a family of weakly orthogonal polynomials which obey a three-te rm recursion relation. in particular, we prove that (normalizable) exa ctly solvable one-dimensional systems are characterized by the fact th at their associated polynomials satisfy a two-term recursion relation. We study the properties of the family of weakly orthogonal polynomial s defined by an arbitrary one-dimensional quasi-exactly solvable Hamil tonian, 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 polynomia l systems. (C) 1996 American Institute of Physics.