Zoran Skoda
Tremblay-Turbiner-Winternitz system

A superintegrable system with 2 degrees of freedom. In polar coordinates, the Hamiltonian is

H k= r 21r r1r 2 ϕ 2+w 2r 2+1r 2(αsin 2kϕ+βcos 2kϕ) H_k = -\partial_r^2 - \frac{1}{r}\partial_r -\frac{1}{r^2}\partial_\phi^2 + w^2 r^2 + \frac{1}{r^2}(\frac{\alpha}{sin^2 k\phi}+\frac{\beta}{cos^2 k\phi} )
  • Frédérick Tremblay, Alexander V. Turbiner, Pavel Winternitz, An infinite family of solvable and integrable quantum systems on a plane, arxiv/0910.0299

quantum version conserved quantities

X k= ϕ 2+k 2(αsin 2kϕ+βcos 2kϕ) X_k = -\partial_\phi^2 + k^2 (\frac{\alpha}{sin^2 k\phi}+\frac{\beta}{cos^2 k\phi} )
Y 2k=...... Y_{2k} = ... ...

where Y 2kY_{2k} is certain diff operator of order 2k2k. For small coupling constants kk, α\alpha, β\beta

cf. dihedral D nD_n-root Calogero model (from Olshanetsky-Perelomov)

  • C. Quesne, Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd kk, arxiv/0911.4404

Possibly open questions: Inequivalent quantizations ? Is any of those compatible with the integrability ?

Created on October 26, 2010 at 09:23:38. See the history of this page for a list of all contributions to it.