Do quasi-exactly solvable systems always correspond to orthogonal polynomials?

Abstract

We consider two quasi-exactly solvable problems in one dimension for which the Schrodinger equation can be converted to Heun's equation. We show that in neither case the Bender-Dunne polynomials form an orthogonal set. Using the anti-isopectral transformation we also discover a new quasi-exactly solvable problem and show that even in this case the polynomials do not form an orthogonal set.