Quasi-Exactly Solvable Potentials on the Line and Orthogonal Polynomials

TL;DR

This paper explores the relationship between quasi-exactly solvable quantum potentials and their associated orthogonal polynomials, revealing new properties of these polynomials and their measures, and characterizing weakly orthogonal systems.

Contribution

It establishes a link between quasi-exactly solvable Hamiltonians and weakly orthogonal polynomials, including their recursion relations and measure support, extending understanding beyond exactly solvable cases.

Findings

Weakly orthogonal polynomials have finite support measures.

Moments grow polynomially and satisfy a linear difference equation.

Exactly solvable systems correspond to two-term recursion relations.

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 $k$-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.