Quasi exact solvability

A linear differential operator L is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions { V } n such that L : { V } n → { V } n , where n is a dimension of { V } n . There are two important cases:

1. { V } n is the space of multivariate polynomials of degree not higher than some integer number; and
2. { V } n is a subspace of a Hilbert space. Sometimes, the functional space { V } n is isomorphic to the finite-dimensional representation space of a Lie algebra g of first-order differential operators. In this case, the operator L is called a g-Lie-algebraic Quasi-Exactly-Solvable operator. Usually, one can indicate basis where L has block-triangular form. If the operator L is of the second order and has the form of the Schrödinger operator, it is called a Quasi-Exactly-Solvable Schrödinger operator.

The most studied cases are one-dimensional s l ( 2 ) -Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian

{ H } = − d 2 d x 2 + a 2 x 6 + 2 a b x 4 + [ b 2 − ( 4 n + 3 + 2 p ) a ] x 2 ,   a ≥ 0   ,   n ∈ N   ,   p = { 0 , 1 } ,

where (n+1) eigenstates of positive (negative) parity can be found algebraically. Their eigenfunctions are of the form

Ψ ( x )   =   x p P n ( x 2 ) e − a x 4 4 − b x 2 2   ,

where P n ( x 2 ) 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.