University of Minnesota
School of Physics & Astronomy

Fine Theoretical Physics Institute Seminar

Monday, December 7th 2009
Speaker: Alexander Turbiner, National University of Mexico, Mexico City
Subject: A new family of planar solvable and integrable Schroedinger equations

It is shown that the Smorodinsky-Winternitz potential, BC_2 rational model, 3-body Calogero model, Wolves potential are all the members of a continuous family of planar solvable and integrable Schroedinger equations marked by some continuous parameter. Their spectra is always linear in quantum numbers. Hidden algebra of the family for integer values of the parameter is uncovered. It is non-semi-simple Lie algebra gl(2)x R^{k+1} realized as vector fields on line bundles over k-Hirzebruch surface. Obtained potential admits quasi-exactly solvable (QES) generalization with the same hidden algebra gl(2) x R^{k+1}. The question about super-integrability of the QES potential remain open yet.Classical-mechanical analogue of the family is presented. It has a property of integrability while the solvability is replaced by a feature that all finite trajectories are closed.

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