A duality of the classical action yields a reflection symmetry of the   quantum energy spectrum
  The concept of duality reflects a link between two seemingly different physical objects. An example in quantum mechanics is a situation where the spectra (or their parts) of two Hamiltonians go into each other under a certain transformation. We term this phenomenon as the energy-spectrum reflection symmetry.   We develop an approach to this class of problems, based on the global properties of the Riemann surface of the quantum momentum function, a natural quantum-mechanical analogue to the classical momentum. In contrast to the algebraic method, which we also briefly review, our treatment provides an explanation to the long-noticed matching of the perturbative and WKB expansions of dual energy levels. Our technique also reveals the classical origins of duality. 
Michael Kreshchuk and Tobias Gulden
math-ph hep-th math.MP quant-ph