  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.
