Superintegrable systems in quantum mechanics and classical Lie theory

Abstract

The relation is established between some concepts of quantum mechanics and those of soliton theory. In particular, superintegrable systems in two-dimensional quantum mechanics are shown to be invariant under generalized Lie symmetries and to allow recursion operators.

Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion

“…[5][6][7] Like much of the later work on superintegrable systems, it was restricted to the case of second-order integrals of motion. [8][9][10][11][12][13] This case turned out to have an intimate connection with the separation of variables in the HamiltonJacobi and Schrödinger equations.…”

Superintegrability in a two-dimensional space of nonconstant curvature

“…All three of the above systems are special cases of the superintegrable systems found in E 2 [4,11]. They were shown to be exactly solvable in Ref.…”

“…More recently a relation between superintegrable systems and generalized Lie symmetries has been established [11], as well as their relation to exactly solvable problems in quantum mechanics [12]. Recently [13,14,15] it has been possible to classify all maximally superintegrable systems for spaces of constant curvature (possibly zero) in two dimensions for which all the extra constants of the motion are at most quadratic in the canonical momenta.…”

Lorentz and Galilei invariance on lattices

“…(22), are examples of generalized Lie symmetries in both discrete and continuous quantum mechanics [37].…”