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We present the higher-dimensional generalization of anisotropic (pseudo)spherical oscillator suggested recently in [arXiv:0710.5001], and the related spherical and pseudosperical generalizations (MICZ-)Kepler like systems with Stark term…
We present some generalization of 16D oscillator by anisotropic and nonlinear inharmonic terms and its dual analog for 9D related MICZ-Kepler systems by generalized version of the Kustaanheimo-Stiefel transformation. The solvability of the…
We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their…
This paper deals with dynamical system that generalizes the MIC-Kepler system. It is shown that the Schr\"{o}dinger equation for this generalized MIC-Kepler system can be separated in spherical and parabolic coordinates. The spectral…
It is shown that the generalized MIC-Kepler system and four-dimensional singular oscillator are dual to each other and the duality transformation is the generalized version of the Kustaanheimo-Stiefel transformation.
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that…
We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and (two-sheet) hyperboloid. We then construct their energy spectra and normalized wave functions, concluding that the suggested systems are minimally…
A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic…
A new integrable generalization to the 2D sphere $S^2$ and to the hyperbolic space $H^2$ of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is…
We propose an exactly-solvable model of the quantum oscillator on the class of K\"ahler spaces (with conic singularities), connected with two-dimensional complex projective spaces. Its energy spectrum is nondegenerate in the orbital quantum…
This article deals with a quantum-mechanical system which generalizes the ordinary isotropic harmonic oscillator system. We give the coefficients connecting the polar and Cartesian bases for D=2 and the coefficients connecting the Cartesian…
We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole…
This article deals with nonrelativistic study of a D-dimensional superintegrable system, which generalizes the ordinary isotropic oscillator system. The coefficients for the expansion between the hyperspherical and Cartesian bases…
The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a $su(2)$ nonAbelian monopole interaction (Yang-Coulomb monopole). This system is also related to a 8D isotropic harmonic oscillator by…
In this paper, we analyze the modification of integrable models in the $\kappa$-deformed space-time. We show that two dimensional isotropic oscillator problem, Kepler problem and MICZ-Kepler problem in $\kappa$-deformed space-time admit…
The Kustaanheimo-Stiefel (KS) transformation maps the non-linear and singular equations of motion of the three-dimensional Kepler problem to the linear and regular equations of a four-dimensional harmonic oscillator. It is used extensively…
A superintegrable generalization of the classical and quantum Zernike systems is reviewed. The corresponding Hamiltonians are endowed with higher-order integrals and can be interpreted as higher-order superintegrable perturbations of the 2D…
We consider two three-dimensional isotropic harmonic oscillators interacting with the quantum electromagnetic field in the Coulomb gauge and within dipole approximation. Using a Bogoliubov-like transformation, we can obtain transformed…
The generalized deformed oscillator schemes introduced as unified frameworks of various deformed oscillators are proved to be equivalent, their unified representation leading to a correspondence between the deformed oscillator and the N=2…
Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to…