Quasi-exact solvability

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

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

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

where 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.


  • A.V. Turbiner and A.G. Ushveridze, Spectral Singularities and the Quasi-Exactly-Solvable Problem, Phys.Lett. 126A, 181-183 (1987).
  • A.V. Turbiner, Quasi-Exactly-Solvable Problems and the algebra, Comm.Math.Phys. 118, 467-474 (1988)
  • 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, MR 1329549

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