arXiv subjects
Publications and source records attributed to Frédérick Tremblay.
We show that all bounded trajectories in the two dimensional classical system with the potential $V(r,ϕ)=ω^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k ϕ}}+ \frac{βk^2}{r^2 \sin^2 {k ϕ}}$ are closed for all integer and rational values of $k$. The period is $T=\fracπ{2ω}$ and does not depend on $k$. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.
An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.