44 problems

Exact-solvability conjecture. All maximally superintegrable systems are exactly solvable.

A quantum system with $n$ degrees of freedom is called maximally superintegrable when it possesses $2n-1$ integrals of motion, the maximum possible number. An exactly solvable quan…

Maximal superintegrability conjecture. The Hamiltonian system with the potentials in equation $$ is maximally superintegrable.

TTW quantum-superintegrability conjecture. The TTW system is quantum superintegrable for all rational values of $k$.

Multiseparability conjecture. Any MS system should separate in multiple coordinate systems.

Superintegrability–exact-solvability conjecture. All maximally superintegrable systems are exactly solvable.

Let $E^n$ be $n$-dimensional Euclidean space, and consider superintegrable systems whose integrals of motion are differential operators of order at most two. The Montreal conjectur…

Let $H_k$ be the Tremblay–Turbiner–Winternitz Hamiltonian with positive integer parameter $k$, let ${\cal Y}_{2k}$ denote a candidate integral of motion, let $\Psi_0$ be the ground…

A quadratically superintegrable system is a superintegrable system whose integrals of motion are differential operators of order at most two; its algebra of integrals is the algebr…

Let $E^n$ denote $n$-dimensional Euclidean space, and consider quantum superintegrable systems on $E^n$ whose integrals of motion have differential order at most two. The Montreal…

A maximally superintegrable quantum system in flat space is a quantum system whose number of functionally independent integrals of motion is maximal possible; an exactly-solvable s…

Let $s$ and $N$ define the monomial matrix model, let $a$ and $\beta$ be its contour and deformation parameters, let $C_{s,a}$ be the integration contour, and let $U_R^{(s,1)}$ be…

Superintegrable models are obtained from reductions of free particles in higher-dimensional spaces. Reduction conjecture. This reduction principle also holds for non-Hermitian Hami…

Consider the classical Hamiltonian family $H^{(cl)}_n$ and its quantum version $H^{(q)}_n$, with parameters as introduced in the paper. The authors distinguish known reductions fro…

For $\gamma=0$, $n=2$, $s=1$, and $\omega_1=\omega_2$, the Hamiltonian family reduces, after suitable polar-coordinate transformations, to the Tremblay–Turbiner–Winternitz system.…

The paper introduces the $n$-dimensional Hamiltonian family … where $\gamma\geq0$, $k\neq0$, $s$, $\alpha_1,\ldots,\alpha_n$, and $\omega_1,\ldots,\omega_n$ are parameters. Superin…

A character bilinear is a product of two characters evaluated on the same or shifted power sums, and an average is taken with respect to a matrix-model or eigenvalue-model measure.…

The order conjecture. For arbitrary $N$, the polynomial algebra above is of $(2N-1)$th order; for $N=2$, it reduces to the well-known cubic Higgs algebra of the proper Zernike syst…

Let $N$ be the order of a superintegrable system, let $L_z$ denote the angular momentum, and let $p_1,p_2$ be the Cartesian momenta. Let $V_1$ and $V_2$ be functions of the Cartesi…

Consider 3-body choreographies with zero total angular momentum arising for pairwise potentials that may be attractive at all distances or repulsive at small distances and attracti…

A Figure-8-shape choreographic motion is a 3-body choreographic trajectory having the Figure-8 shape discovered for 3-body dynamics. A 3-body potential theory is superintegrable al…

A maximally superintegrable classical system is a classical system possessing the maximal number of functionally independent integrals of motion. The superintegrability conjecture.…

A maximally superintegrable system is a classical dynamical system possessing the maximal number of functionally independent integrals of motion. Quantum exact-solvability conjectu…

Exact-solvability conjecture. All maximally superintegrable quantum systems in Euclidean spaces are exactly solvable.

Consider a Hamiltonian system in two-dimensional Euclidean space that separates in orthogonal coordinates and admits a third-order integral of motion. In the parabolic case, the kn…