Quantum Parabolic Sombrero

We have discussed the energy levels and probability distribution density for a quantum particle placed in the two-dimensional sombrero-shaped potential $V(\rho,\rho_0)=\mu\omega^2|\rho^2-\rho_0^2|/2$.Comment: 10 pages, LaTex, 6 figures (eps). accepted in Phys. Lett.

Quantum Systems with Hidden Symmetry. Interbasis Expansions

This monograph is the English version of the book "Quantum systems with hidden symmetry. Interbasis expansions" published in 2006 by the publishing house FIZMATLIT (Moscow) in Russian. When compiling this version of the book, typos and inaccuracies noted since the release of the Russian edition have been corrected

All-particle primary energy spectrum in the 3-200 PeV energy range

We present all-particle primary cosmic-ray energy spectrum in the 3-200 PeV energy range obtained by a multi-parametric event-by-event evaluation of the primary energy. The results are obtained on the basis of an expanded EAS data set detected at mountain level (700 g/cm^2) by the GAMMA experiment. The energy evaluation method has been developed using the EAS simulation with the SIBYLL interaction model taking into account the response of GAMMA detectors and reconstruction uncertainties of EAS parameters. Nearly unbiased (<5%) energy estimations regardless of a primary nuclear mass with an accuracy of about 15-10% in the 3-200 PeV energy range respectively are attained. An irregularity ('bump') in the spectrum is observed at primary energies of ~74 PeV. This bump exceeds a smooth power-law fit to the data by about 4 standard deviations. Not rejecting stochastic nature of the bump completely, we examined the systematic uncertainties of our methods and conclude that they cannot be responsible for the observed feature.Comment: Accepted by J.Phys.G: Nucl.Part.Phy

Quantum oscillator as 1D anyon

It is shown that in one spatial dimension the quantum oscillator is dual to the charged particle situated in the field described by the superposition of Coulomb and Calogero-Sutherland potentials.Comment: 9 pages, LaTe

Second Hopf map and Yang-Coulomb system on 5d (pseudo)sphere

Using the second Hopf map, we perform the reduction of the eight-dimensional (pseudo)spherical (Higgs)oscillator to a five-dimensional system interacting with a Yang monopole. Then, using a standard trick, we obtain, from the latter system, the pseudospherical and spherical generalizations of the Yang-Coulomb system (the five dimensional analog of MICZ-Kepler system). We present the whole set of its constants of motions, including the hidden symmetry generators given by the analog of Runge-Lenz vector. In the same way, starting from the eight-dimensional anisotropic inharmonic Higgs oscillator, we construct the integrable (pseudo)spherical generalization of the Yang-Coulomb system with the Stark term.Comment: 10 pages, PACS: 03.65.-w, 02.30.Ik, 14.80.H

3D Oscillator and Coulomb Systems reduced from Kahler spaces

We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kahler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kahler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid are originated. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to monopole, and find their four-dimensional origins. In the latter case the metric of configuration space is non-Kahler one. Finally, we extend these results to the family of Kahler spaces with conic singularities.Comment: To the memory of Professor Valery Ter-Antonyan, 11 page

Anisotropic inharmonic Higgs oscillator and related (MICZ-)Kepler-like systems

We propose the integrable (pseudo)spherical generalization of the four-dimensional anisotropic oscillator with additional nonlinear potential. Performing its Kustaanheimo-Stiefel transformation we then obtain the pseudospherical generalization of the MICZ-Kepler system with linear and $\cos\theta$ potential terms. We also present the generalization of the parabolic coordinates, in which this system admits the separation of variables. Finally, we get the spherical analog of the presented MICZ-Kepler-like system.Comment: 7 page