21 problems

The quantization of a classical superintegrable system can produce a quantum superintegrable system whose integrals of motion generate a quadratic associative algebra; the correspo…

New-transcendent conjecture. The general solutions of these equations cannot be expressed in terms of the six Painlevé transcendents; hence the equations define a new transcendent.

Let $k$ be an integer. For the TTW quantum system, let $H$ be the Hamiltonian, let ${\cal I}_{1}$ and ${\cal I}_{2}$ be its two integrals, and set … The polynomial algebra of integ…

Let $k$ be an integer, and consider the TTW quantum system on the plane with Hamiltonian $H$ and two integrals ${\cal I}_{1}$ and ${\cal I}_{2}$, together with their commutator…

Let $R$, $a_3$, and $b_1$ be the integral and parameters appearing in the superintegrable system, and let $\bar{\psi}_n$ be the states generated from the ground state by … For each…

Linearity conjecture. Superintegrable systems in two-dimensional non-Euclidean space can be reduced to linear equations by means of their hidden symmetries.

Second-family conjecture. A second infinite family of doubly exotic quantum superintegrable potentials occurs, with

First-family conjecture. An infinite family of doubly exotic quantum superintegrable potentials occurs, with

A Zoll metric of revolution is a metric of revolution all of whose geodesics are closed. A superintegrable (SI) system is an integrable Hamiltonian system admitting the additional…

The construction begins with independent one-dimensional algebraic systems in the $x$- and $y$-spaces, each consisting of a natural Hamiltonian and a polynomial integral; combining…

Consider quantum superintegrable Hamiltonian systems in the Euclidean plane, with polar potential $V(r,\theta)=R(r)+S(\theta)/r^2$, and let $Y_I^{(N)}$ and $Y_{II}^{(N)}$ denote th…

Consider classical superintegrable Hamiltonian systems in the Euclidean plane, expressed in polar coordinates with potential $V(r,\theta)=R(r)+S(\theta)/r^2$. Let $Y_I^{(N)}$ and…

A superintegrable system is a Hamiltonian system possessing the maximal number of functionally independent integrals of motion, with at least one integral polynomial in the momenta…

Superintegrable systems are dynamical systems possessing sufficiently many integrals of motion; in the context of this paper, the claim concerns such systems and their associated s…

Higher-rank polynomial algebra conjecture. Symmetry algebras of $n$-dimensional superintegrable systems would in general be higher-rank polynomial algebras, although only a few iso…

In the Euclidean plane $E_2$, consider superintegrable systems with one integral of motion of order $N\geq 3$ and two other integrals of order at most $2$. Painlevé-property exotic…

The Askey scheme consists of families of hypergeometric orthogonal polynomials, and second order superintegrable systems in dimension two are related to these families through limi…

Linearizability conjecture. All classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.

A superintegrable system is a Hamiltonian system that admits more integrals of motion than degrees of freedom; here the Hamiltonian is … with two additional integrals polynomial in…

The generic superintegrable system on the 3-sphere has a quadratic algebra generated by the two-variable Wilson polynomials, and limiting processes are expected to produce the quad…

Explicit polynomial integral conjecture. The extra polynomial first integral for this system is given, up to the factor $(-1)^n$, by