Higher Order Quantum Superintegrability: A New “Painlevé Conjecture”

Abstract

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order N > 2. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For N = 3, 4 and 5 these equations have the Painlevé property. We conjecture that this is true for all N ≥ 3. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any N ) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

Superintegrability of three-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems

“…Until recently, most studies on superintegrability were concentrated on the quadratic case but in these last years the existence of systems possessing integrals of motion of higher-order in the momenta (not arising from separability) has also been studied [31]- [37] but mainly in the twodimensional Euclidean space. This paper is devoted to the study of some superintegrable systems on 3-dimensional conformally Euclidean spaces (see [38]- [46] for papers on this particular geometry).…”

Symmetries of the Schroedinger-Pauli equations for charged particles and quasirelativistic Schroedinger equations

“…where ε bcd is the absolutely antisymmetric unit tensor. Substituting (17) into (19) and using definition (7) we reduce the latter equation to the following form:…”

“…The second (and higher) order symmetries can be related to such nice properties of SE as superintegrability and supersymmetry, see surveys [5] and [6]. We will not discuss them here but mention that searching for such symmetries is still a very popular business, and the modern trends in this field are related to the third order and even arbitrary order integrals of motion [7,8,9], see also paper [10] where the determining equations for such symmetries were presented.…”

Superintegrable systems with spin and second-order (pseudo)tensor integrals of motion

“…More recently, exotic and standard potentials appearing in classical and quantum superintegrable systems have been studied both in Cartesian and polar coordinates [51][52][53][54][55][56][57][58][59][60].…”