A class of spectral problems with a hidden Lie-algebraic structure is considered. We define a duality transformation which maps the spectrum of one quasi-exactly solvable (QES) periodic potential to that of another QES periodic potential. The self-dual point of this transformation corresponds to the energy-reflection symmetry found previously for certain QES systems. The duality transformation interchanges bands at the bottom (top) of the spectrum of one potential with gaps at the top (bottom) of the spectrum of the other, dual, potential. Thus, the duality transformation provides an exact mapping between the weak coupling (perturbative) and semiclassical (nonperturbative) sectors.
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We thank C. Bender, A. Turbiner, A. Vainshtein, and M. Voloshin for valuable discussions, and T. ter Veldhuis and M. Voloshin for assistance with Mathematica. G.D. is supported in part by DOE Grant DE-FG02-92ER40716, and M.S. is supported in part by DOE Grant DE-FG02-94ER408.