Quasi-exact solvability
Artemio González-López, Niky Kamran, Peter J. Olver · Contemporary mathematics - American Mathematical Society · 1994
. This paper surveys recent work on quasi-exactly solvable Schrodinger operators and Lie algebras of differential operators. 1. Introduction. Lie algebraic and Lie group theoretic methods have played a significant role in the development of quantum mechanics since its inception. In the classical applications, the Lie group appears as a symmetry group of the Hamiltonian operator, and the associated representation theory provides an algebraic means for computing the spectrum. Of particular importance are the exactly solvable problems, such as the harmonic oscillator or the hydrogen atom, whose point spectrum can be completely determined using purely algebraic methods. The fundamental concept of a "spectrum generating algebra" was introduced by Arima and Iachello, [4], [5], to study nuclear physics, and subsequently, by Iachello, Alhassid, Gursey, Levine, Wu and their collaborators, was also successfully applied to molecular dynamics and spectroscopy, [19], [22], and scattering theory, [...