Superintegrability in a two-dimensional space of nonconstant curvature

Abstract

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of twodimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical * Email: e.kalnins@waikato.ac.nz † Email: jonathan@math.waikato.ac.nz ‡ Email: wintern@crm.umontreal.ca 1 momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton-Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.2

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

“…Four classes of them exist in E 2 , 5 maximally superintegrable (2n − 1 = 5 operators commuting with H) and 8 minimally superintegrable ones (n + 1 = 4 operators) in E 3 . These results have been recently extended to two and three dimensional spaces of constant curvature and to complex spaces [17][18][19][20] and also to certain two dimensional spaces of nonconstant curvature 21 .…”

Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations

“…The explicit solution of the geodesic flows for all these spaces has been studied in [2], as well as a method to introduce (super)integrable potentials on them by adding a potential term of the form U (zJ (2) − ); in this way, some known potentials are recovered (appearing in the classifications [6,7]) and also new ones are obtained. We recall that another approach to superintegrability on 2D spaces of variable curvature can be found in [8,9].…”

“…Likewise, their "deformed" counterpart (understood as spaces with non-constant curvature) have been deduced from the integrable metric (8). The explicit solution of the geodesic flows for all these spaces has been studied in [2], as well as a method to introduce (super)integrable potentials on them by adding a potential term of the form U (zJ (2) − ); in this way, some known potentials are recovered (appearing in the classifications [6,7]) and also new ones are obtained.…”