Doubly Exotic Nth-Order Superintegrable Classical Systems Separating in Cartesian Coordinates

Abstract

Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space E 2 are explored. The study is restricted to Hamiltonians allowing separation of variables V (x, y) = V 1 (x) + V 2 (y) in Cartesian coordinates. In particular, the Hamiltonian H admits a polynomial integral of order N > 2. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of the formalism are non-linear compatibility conditions and the algebra of the integrals of motion. The case N = 5, where doubly exotic confining potentials appear for the first time, is completely solved to illustrate the present approach. The general case N > 2 and a formulation of inverse problem in superintegrability are briefly discussed as well.

Superintegrable and Scale-Invariant Quantum Systems with Position-Dependent Mass

“…Evaluating the commutator in (17) and equating to zero the coefficients for the linearly independent differential operators ∂ a ∂ b ∂ c and ∂ a we come to the following determining equations (see Appendix)…”

“…The systematic study of them had been started with seminal papers [9] were the competed classification of the second order integrals of motion for the 2d Schrödinger equation was proposed. These results induced a great many of generalizations, staring with 3d models [10], [11] and continuing with models including matrix potentials [12,13,14] and the higher (and even arbitrary) order integrals of motion [15,16,17,18]. Moreover, such generalizations include the systems more generic than the standard Schrödinger equations, namely, Schrödinger equations with position dependent mass (PDM).…”

“…To deduce determining equations ( 18)- (20) it is sufficient to evaluate the commutator in (17) and equate the coefficients for the linearly independent differential operators. We present here the routine calculations requested to achieve this goal since the form of the determining equations strongly depends on the chosen representation of the Hamiltonian (compare (2) and ( 3) and the integral of motion.…”

Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups

“…The systematic study of them had been started with seminal papers [9] were the competed classification of the second order integrals of motion for the 2d Schrödinger equation was proposed. These results induced a great many of generalizations, starting with 3d models [10,11] and continuing with models including matrix potentials [12][13][14] and the higher (and even arbitrary) order integrals of motion [15][16][17][18]. Moreover, such generalizations include the systems more generic than the standard Schrödinger equations, namely, Schrödinger equations with position dependent mass (PDM).…”

“…The modern trends are to study the 2d superintegrable systems admitting integrals of motion of the third and even arbitrary orders [16,17]. However, there is only a particular progress in this direction which is restricted to the systems with constant masses and very specific kind of potentials.…”