NEW QUASI-EXACTLY SOLVABLE HAMILTONIANS IN 2 DIMENSIONS

Quasi-exactly solvable Schrodinger operators have the remarkable prope rty that a part of their spectrum can be computed by algebraic methods . Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differential operators - the ''hidden symm etry algebra.'' In this paper we develop some general techniques for c onstructing quasi-exactly solvable operators. Our methods are applied to provide a wide variety of new explicit two-dimensional examples (on both flat and curved spaces) of quasi-exactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algeb ras.