Quasi-Exactly Solvable Systems and Orthogonal Polynomials

TL;DR

This paper establishes a connection between quasi-exactly solvable quantum models and orthogonal polynomials, showing how wave functions generate these polynomials and how their properties determine solvable energy levels.

Contribution

It reveals a novel correspondence linking quasi-exact solvability in quantum mechanics to orthogonal polynomial systems, providing a new perspective on spectral analysis.

Findings

Wave functions generate orthogonal polynomials in energy.

Vanishing polynomial norms indicate quasi-exact solvability.

Zeros of critical polynomials give quasi-exact energy eigenvalues.

Abstract

This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials $\{ P_n\}$. The quantum-mechanical wave function is the generating function for the $P_n (E)$, which are polynomials in the energy $E$. The condition of quasi-exact solvability is reflected in the vanishing of the norm of all polynomials whose index $n$ exceeds a critical value $J$. The zeros of the critical polynomial $P_J(E)$ are the quasi-exact energy eigenvalues of the system.