Zoran Skoda Tremblay-Turbiner-Winternitz system

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

$$H_k = -\partial_r^2 - \frac{1}{r}\partial_r -\frac{1}{r^2}\partial_\phi^2 + w^2 r^2 + \frac{1}{r^2}(\frac{\alpha}{sin^2 k\phi}+\frac{\beta}{cos^2 k\phi} )$$

• 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

$$X_k = -\partial_\phi^2 + k^2 (\frac{\alpha}{sin^2 k\phi}+\frac{\beta}{cos^2 k\phi} )$$

$$Y_{2k} = ... ...$$

where $Y_{2k}$ is certain diff operator of order $2k$. For small coupling constants $k$, $\alpha$, $\beta$…

cf. dihedral $D_n$-root Calogero model (from Olshanetsky-Perelomov)

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

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