Runge-Lenz vector and Stark effect in Calogero-Coulomb problem

by Armen Nersessian (Yerevan State Univ.)

Wednesday 3 Jun 2015, 14:30 → 16:00 Europe/Rome

Aula Seminari (LNF)

We show that deformation N-dimensional (Euclidean, spherical and hyperbolic) Coulomb system by the patential of rational Calogero model preserves superintegrability property of Coulomb problem. Then we find explicit expression of the Runge-Lenz vector and symmetry algebra of rational Calogero-Coulomb problem, formulated in terms of Dunkl operators. We find that they are proper deformations of their Coulomb counterparts. This observation permits to claim that most of properties of Coulomb and oscillator systems can be lifted to their Calogero-extended analogs by the proper replacement of momenta by Dunkl momenta operators. Particularly , we show that N-dimensional Calogero-Coulomb system preserves integrability property in the presence of Stark term, and construct its complete set of constants of motion. We find, that in parabolic coordinates the systems admits complete separation of variables for N=2,3 and partial ones for $N>3$. The system possesses linear Stark effect and find its explicit expression.Finally, we show, that two-center Calodjero-Coulomb problem is also integrable systems, which admits, in elliptic coordinates, the complet separation of variables for N=2,3 and partial ones for N>3.