One-dimensional quasi-exactly solvable Schrodinger equations

Quasi-Exactly Solvable Schrodinger Equations occupy an intermediate place between exactly-solvable (e.g. the harmonic oscillator and Coulomb problems, etc.) and non solvable ones. Mainly, they were discovered in the 1980s. Their major property is an explicit knowledge of several eigenstates while the remaining ones are unknown. Many of these problems are of the anharmonic oscillator type with a special type of anharmonicity. The Hamiltonians of quasi-exactly-solvable problems are characterized by the existence of a hidden algebraic structure but do not have any hidden symmetry properties. In particular, all known one-dimensional (quasi)-exactly-solvable problems possess a hidden sl[(2, R)-Lie algebra. They are equivalent to the sl(2, R) Euler-Arnold quantum top in a constant magnetic field. Quasi-Exactly Solvable problems are highly non-trivial, they shed light on the delicate analytic properties of the Schrodinger Equations in coupling constant, they lead to a non-trivial class of potentials with the property of Energy-Reflection Symmetry. The Lie algebraic formalism allows us to make a link between the Schrodinger Equations and finite difference equations on uniform and/or exponential lattices, it implies that the spectra is preserved. This link takes the form of quantum canonical transformation. The corresponding isospectral problems for finite-difference operators are described. The underlying Fock space formalism giving rise to this correspondence is uncovered. For a quite general class of perturbations of unperturbed problems with the hidden Lie algebra property we can construct an algebraic perturbation theory, where the wavefunction corrections are of polynomial nature, thus, can be found by algebraic means. In general, Quasi-Exact-Solvability points to the existence of a hidden algebra formalism which ranges from quantum mechanics to 2-dimensional conformal field theories. (C) 2016 Elsevier B.V. All rights reserved.