Quasi-Exact Solvability

Quasi-exact solvability is the property of a linear differential operator that preserves an explicitly constructible finite-dimensional invariant subspace of functions, allowing part of its eigenvalue problem to be solved by finite matrix algebra (Turbiner 1988, 2016). The basis must be known explicitly, rather than defined in terms of unknown eigenfunctions. For a Schrödinger operator, the resulting functions must also satisfy its boundary conditions and be square-integrable to give eigenfunctions.

A standard construction uses the vector space of polynomials

(1)

where is a nonnegative integer. If , restricting to this invariant subspace gives an matrix. Its characteristic polynomial determines the corresponding eigenvalues.

For example, consider the Schrödinger operator

(2)

on the real line. The substitution

(3)

turns into , where and

(4)

For an integer with , the only term in that can raise the polynomial degree is . It vanishes at , proving that preserves . This gives square-integrable even eigenfunctions. For , the matrix in the basis is

(5)

so the two eigenvalues are , with . At suitable parameter values, sextic potentials with additional quartic and quadratic terms also have finite solvable subsets (Child et al. 2000, Turbiner 2016).