D-dimensional massless particle with extended gauge invariance

We propose the model ofD-dimensional massless particle whose Lagrangian is given by theN-th extrinsic curvature of world-line. The system hasN+1 gauge degrees of freedom constitutingW-like algebra; the classical trajectories of the model are space-like curves which obey the conditionsk _N+a=k_N−a, k_2N =0,a=1, ...,N−1,N≤[(D−2)/2], while the firstN curvaturesk _i remain arbitrary. We show that the model admits consistent formulation on the anti-DeSitter space. The solutions of the system are the massless irreducible representations of Poincaré group withN nonzero helicities, which are equal to each other. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Uniform estimates in the Poincaré-Aronszajn theorem on the separation of singularities of analytic functions

We study the possibility of splitting any bounded analytic function ƒ with singularities in a closed set E ⋃ F as a sum of two bounded analytic functions with singularities in E and F respectively. We obtain some results under geometric restrictions on the sets E and F and we provide some examples showing the sharpness of the positive results. The first author is partially supported by RFBR grant #06-01-00313 and by the grant for Leading Scientific Schools #NSH-2266-2003.1. The last author is supported by DGICYT grant MTM2005-08984-C02-02 and the CIRIT grant 2005SGR00611.     

Hidden symmetries of integrable conformal mechanical systems

We split the generic conformal mechanical system into a “radial” and an “angular” part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N=4 supersymmetric “angular” Hamiltonian system one may construct a new system with full N=4 superconformal D(1,2;α) symmetry.     

Hyperboloid,instanton, oscillator

We suggest the exactly solvable model of the oscillator on a four-dimensional hyperboloid which interacts with a SU(2)$\mathit{SU}(2)$ instanton. We calculate its wavefunctions and spectrum.     

Lobachevsky geometry of (super)conformal mechanics

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) . We also extend this picture to the N=2k superconformal mechanics described in terms of Lobachevsky superspace .     

Integrability of Geodesics in Near-Horizon Extremal Vanishing Horizon Myers–Perry Black Holes

Physics of Atomic Nuclei - We study geodesics on the near horizon geometry of odd-dimensional Myers–Perry black hole in the extremal vanishing horizon (EVH) limit. Knowing the fact that...     

How to relate the oscillator and Coulomb systems on spheres and pseudospheres?

Physics of Atomic Nuclei - We show that oscillators on a sphere and a pseudosphere are related, by the so-called Bohlin transformation, with Coulomb systems on a pseudosphere: even states of an...     

Lobachevsky geometry in TTW and PW systems

Physics of Atomic Nuclei - We review the classical properties of Tremblay–Turbiner–Winternitz and Post–Wintenitz systems and their relation with N-dimensional rational Calogero...     

Accelerating the convergence of trigonometric series

A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.