Quasi-exact solvability

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

1. ${\displaystyle \{{\mathcal {V}}\}_{n}}$ is the space of multivariate polynomials of degree not higher than some integer number; and
2. ${\displaystyle \{{\mathcal {V}}\}_{n}}$ is a subspace of a Hilbert space. Sometimes, the functional space ${\displaystyle \{{\mathcal {V}}\}_{n}}$ 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 ${\displaystyle sl(2)}$-Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian

${\displaystyle \{{\mathcal {H}}\}=-{\frac {d^{2}}{dx^{2}}}+a^{2}x^{6}+2abx^{4}+[b^{2}-(4n+3+2p)a]x^{2},\ a\geq 0\ ,\ n\in \mathbb {N} \ ,\ p=\{0,1\},}$

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

${\displaystyle \Psi (x)\ =\ x^{p}P_{n}(x^{2})e^{-{\frac {ax^{4}}{4}}-{\frac {bx^{2}}{2}}}\ ,}$

where ${\displaystyle P_{n}(x^{2})}$ 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.