A linear differential operator L is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions
{
}
{\displaystyle \{{\mathcal {V}}\}_{n}}
: {
}
→ {
{\displaystyle L:\{{\mathcal {V}}\}_{n}\rightarrow \{{\mathcal {V}}\}_{n},}
where n is a dimension of
There are two important cases: The most studied cases are one-dimensional
{\displaystyle sl(2)}
-Lie-algebraic quasi-exactly-solvable (Schrödinger) operators.
The best known example is the sextic QES anharmonic oscillator with the Hamiltonian
where (n+1) eigenstates of positive (negative) parity can be found algebraically.
Their eigenfunctions are of the form
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.