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 $L:\{𝒱{\}}_n\to \{𝒱{\}}_n,$ 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 $sl(2)$-Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian

$\{\mathscr{H}\}=-\frac{d^2}{d{x}^2}+a^2{x}^6+2ab{x}^4+[b^2-(4n+3+2p)a]x^2,a\ge 0,n\in \mathbb{N},p=\{0,1\},$

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

$\Psi (x)=x^p{P}_n(x^2)e^{-\frac{a{x}^4}{4}-\frac{b{x}^2}{2}},$

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.