Infinite order symmetries for two-dimensional separable Schrödinger equations

Consider a non-relativistic Hamiltonian operator H in 2 dimensions consisting of a kinetic energy term plus a potential. We show that if the associated Schrödinger eigenvalue equation admits an orthogonal separation of variables, there is a calculus to describe the (in general) infinite-order differential operator symmetries of the Schrödinger equation. The calculus is formal but can be made rigorous when all functions in the eigenvaue equation are analytic. The infinite-order calculus exhibits structure that is not apparent when one studies only finite-order symmetries. The search for finite-order symmetries can then be reposed as one of looking for solutions of a coupled system of PDEs that are polynomial in certain parameters. We go further and extend the calculus to the situation where the Schrödinger equation admits a second-order symmetry operator, not necessarily associated with orthogonal separable coordinates

Superintegrability in three-dimensional Euclidean space

Potentials for which the corresponding Schrödinger equation is maximally superintegrable in three-dimensional Euclidean space are studied. The quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

Harmonic Oscillator on the SO(2,2) Hyperboloid

In the present work the classical problem of harmonic oscillator in the hyperbolic space H²₂: z²₀+z²₁−z²₂−z²₃=R² has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on H²₂, as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones

The Kepler-Coulomb problem on SO(2, 2) hyperboloid

In this note the Kepler-Coulomb problem in hyperbolic space H2 2: z0 2 + z1 2 - z2 2 - z3 2 = R2 is discussed. � 2012 Pleiades Publishing, Ltd

Lie-algebra contractions and separation of variables. Three-dimensional sphere

The Inönü-Wigner contraction from the SO(4) group to the Euclidean E(3) group is used to relate the separation of variables in Helmholtz equations for two corresponding homogeneous spaces. We show how the six systems of coordinates on the three-dimensional sphere contracted to nine systems of coordinates on Euclidean space. As a consequence of the Inönü-Wigner contraction we also consider contractions of the integrals of motion. 2009 Pleiades Publishing, Ltd

Classical Kepler-Coulomb problem on SO(2, 2) hyperboloid

In the present work, the problem of the motion of the classical particle in the Kepler-Coulomb field in three-dimensional hyperbolic space H 2 2: z 2 0 + z 2 1 - z 2 2 - z 2 3 = R 2 is solved in the framework of Hamilton-Jacobi equation. The requirements for the existence of bounded motion of particle are formulated. The equation of the trajectory of particle is obtained, and it is shown that all the finite trajectories are closed. It is also demonstrated that under the certain values (zero or negative) of the separation constant A the fall of the particle onto the center takes place. © 2013 Pleiades Publishing, Ltd

Cloud forest dynamics in the mexican neotropics during the last 1300 years

In the present work, the problem of the motion of the classical particle in the Kepler-Coulomb field in three-dimensional hyperbolic space H 2 2: z 2 0 + z 2 1 - z 2 2 - z 2 3 = R 2 is solved in the framework of Hamilton-Jacobi equation. The requirements for the existence of bounded motion of particle are formulated. The equation of the trajectory of particle is obtained, and it is shown that all the finite trajectories are closed. It is also demonstrated that under the certain values (zero or negative) of the separation constant A the fall of the particle onto the center takes place. " 2013 Pleiades Publishing, Ltd.",,,,,,"10.1134/S1063778813090135",,,"http://hdl.handle.net/20.500.12104/40096","http://www.scopus.com/inward/record.url?eid=2-s2.0-84885830563&partnerID=40&md5=52f938ac91d5b20ba1a39a702f821b67",,,,,,"10",,"Physics of Atomic Nuclei",,"127

Two-dimensional imaginary lobachevsky space. Separation of variables and contractions

The In�n�-Wigner contraction from the SO(2, 1) group to the E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the corresponding two-dimensional homogeneous spaces: two-dimensional one sheeted hyperboloid and two-dimensional pseudo-Euclidean space. Here we consider the contraction limits of some basis functions for the subgroup coordinates only. � 2011 Pleiades Publishing, Ltd

Lie algebra contractions on two-dimensional hyperboloid

The Inönü-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E2 and eight on E1,1. The text was submitted by the authors in English. © 2010 Pleiades Publishing, Ltd