A linear differential operator "L" is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions formula_1 such that formula_2 where "n" is a dimension of formula_1. There are two important cases:
The most studied cases are one-dimensional formula_7-Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian
formula_8
where ("n+1") eigenstates of positive (negative) parity can be found algebraically. Their eigenfunctions are of the form
formula_9
where formula_10 is a polynomial of degree "n" and (energies) eigenvalues are roots of an algebraic equation of degree ("n+1"). In general, twelve families of one-dimensional QES problems are known, two of them characterized by elliptic potentials.
References.
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