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

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

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

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

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.

References

• Turbiner, A.V.; Ushveridze, A.G. (1987). "Spectral singularities and quasi-exactly solvable quantal problem". Physics Letters A (Elsevier BV) 126 (3): 181–183. doi:10.1016/0375-9601(87)90456-7. ISSN 0375-9601. Bibcode: 1987PhLA..126..181T. 
• Turbiner, A. V. (1988). "Quasi-exactly-solvable problems and $sl(2,R)$ algebra". Communications in Mathematical Physics (Springer Science and Business Media LLC) 118 (3): 467–474. doi:10.1007/bf01466727. ISSN 0010-3616. Bibcode: 1988CMaPh.118..467T. 
• González-López, Artemio; Kamran, Niky; Olver, Peter J. (1994), "Quasi-exact solvability", Lie algebras, cohomology, and new applications to quantum mechanics (Springfield, MO, 1992), Contemp. Math., 160, Providence, RI: Amer. Math. Soc., pp. 113–140 
• Turbiner, A.V. (1996), "Quasi-exactly-solvable differential equations", in Ibragimov, N.H., CRC Handbook of Lie Group Analysis of Differential Equations, 3, Boca Raton, Fl.: CRC Press, pp. 329–364, ISBN 978-0849394195 
• Ushveridze, Alexander G. (1994), Quasi-exactly solvable models in quantum mechanics, Bristol: Institute of Physics Publishing, ISBN 0-7503-0266-6 

External links

• Olver, Peter, A Quasi-Exactly Solvable Travel Guide, http://math.umn.edu/~olver/q_/qes21.pdf