# Mathematics Colloquia and Seminars

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### A new family of planar solvable and integrable Schroedinger operators

**Mathematical Physics & Probability**

Speaker: | Prof. Alexander Turbiner, Nuclear Science Institute, UNAM, Mexico City |

Location: | 1147 MSB |

Start time: | Wed, Feb 13 2013, 4:10PM |

It is shown that the Smorodinsky-Winternitz potential, $BC_2$ rational model, 3-body Calogero model and Wolfes potential $(G_2)$ are all the members of a continuous family of planar solvable and (super)integrable Schroedinger operators marked by some parameter $k$. There exist two integrals of degree 2 and $2(p+q-1)$ (for rational $k=p/q$). Their spectra is always linear in quantum numbers with degeneracy which depend on $k$. Hidden algebra of the family for integer values of $k$ is uncovered: It is a new infinite-dimensional finitely-generated algebra with unusual Gauss decomposition diagram. It contains as a subalgebra the non-semi-simple Lie algebra $gl(2) \ltimes R^{k+1}$ realized as vector fields on line bundles over k-Hirzebruch surface. Obtained potential admits quasi-exactly solvable (QES) generalization with the hidden algebra $gl(2) \ltimes R^{k+1}$. A particular super-integrability of the QES Hamiltonian is explained. Classical-mechanical analogue of the family is presented. A property of (super)integrability is preserved while the solvability is replaced by a feature that all finite trajectories are closed and isochronous.