Completeness of superintegrability in two-dimensional constant-curvature spaces

Abstract

We classify the Hamiltonians H = p 2 x + p 2 y + V (x, y) of all classical superintegrable systems in two dimensional complex Euclidean space with second-order constants of the motion. We similarly classify the superintegrable Hamiltonians H = J 2 1 + J 2 2 + J 2 3 + V (x, y, z) on the complex 2-sphere where x 2 + y 2 + z 2 = 1. This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

“…In earlier work we have classified the possible superintegrable systems on 2D complex flat space, the two-sphere, and on Darboux spaces. 44,45,[34][35][36] . The theory we present here applies to all 2D spaces and adds greater understanding of the structure of these systems.…”

“…15 Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as a few other spaces. [33][34][35][36][37][38] Here, rather than focus on particular spaces and systems, we employ a theoretical method based on integrability conditions to derive structure common to all such systems. In this paper we consider classical superintegrable systems on a general two-dimensional ͑2D͒ Riemannian manifold, real or complex, and uncover their common structure.…”

Quantum superintegrability and exact solvability in n dimensions

“…Similarly as in the case of potentials containing the Coulomb atom as a special case in 2 dimensions, it is the system (33) (rather than the original system (20)) that is exactly solvable in the sense defined above. Indeed, let us gauge rotate the operators Q 0 , Q 1 and Q 2 and transform to the variables s, t and z.…”

Superintegrability in a two-dimensional space of nonconstant curvature

“…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.…”