Classical and quantum duality of quasi-exactly solvable problems

Abstract

The concept of duality reflects a link between two seemingly different physical objects. An example from quantum mechanics is a situation where the spectra (or their parts) of two different Hamiltonians go into one another under a certain transformation.In this paper we resolve a long-noticed puzzle of matching between the perturbative and WKB expansions of dual energy levels in such potentials. Our approach to this class of problems is based on studying the global properties of the Riemann surface of the quantum momentum function, a natural quantum-mechanical analogue to the classical momentum. Being based on the position representation of the Schrödinger equation and the analytic properties of the wave functions, it explains the matching of the WKB and perturbative series to all orders. Our approach also reveals the classical origins of duality.