15 problems

Arbitrary-parameter GAL relation conjecture. Relations $(2.17)$ and $(2.18)$ are valid whenever either $a+b+f+g$ or one of these modified parameter sums is a non-negative even inte…

Odd-sum GAL spectral-correspondence conjecture. The QES mid-band energy values of the displayed transformed potential are the same as the band-edge eigenvalues of $[a,b,f,g]$.

GAL simultaneous-relation conjecture. Regardless of whether $a+b+f+g$ is odd or even, relations $(2.17)$ and $(2.18)$ are both valid.

Half-integer GAL state-count conjecture. This potential has $k+l+n+3$ QES mid-band states.

Half-integer GAL partner-relation conjecture. Relations $(2.17)$ and $(2.18)$ should hold simultaneously whenever at least one of the four parameters $a,b,f,g$ is a half-integer.

GAL integer-to-half-integer correspondence conjecture. Relations $(2.17)$ and $(2.18)$ should hold simultaneously when $a,b,f,g$ are integers, and the band-edge eigenvalues of the…

The radial Dirac equation is the two-component system for radial functions $f(r)$ and $g(r)$ with energy $E$, mass $M$, centrifugal term $U(r)=\kappa/r$, electrostatic potential…

The models under consideration are classical integrable models obtained from quasi-Yang-Baxter algebras, with action-angle variables as their classical inverse-scattering data. Ush…

Polynomial-space classification conjecture. If $T_k(x,y)$ preserves a finite-dimensional space of all polynomials other than ${\cal P}_{n}^{I}$, ${\cal P}_{n,m}^{II}$, and…

Quasi-exact solvability conjecture. At least a certain class of Inozemtsev models can be made quasi-exactly solvable.

Let $m>1$, and consider the system of linear equations in the unknowns $a_0,a_1,\ldots,a_m$ given by Eqs. (1)--(5) in the source, with parameters and notation as defined there. Con…

Superintegrability–quasi-exact solvability conjecture. All quasi-exactly solvable systems arise from superintegrable systems.

Let $P_J(E,\lambda)$ be the Bender–Dunne polynomial associated with the quasi-exactly solvable sector indexed by $J$, and consider its zeros as functions of $\lambda$ after squarin…

Integer-energy algebraicity conjecture. The only integer energies of this chain are $0$, $1$, $2$, and $3$, and all eigenstates corresponding to these energies are algebraic:

Nonalgebraic-eigenfunction conjecture. Some eigenfunctions of these models are not the ground state times a symmetric function, and their corresponding eigenvalues are not natural…