19 problems

A difference operator is a finite-order expression … acting on functions on $\mathbb Z$, where $T$ is the shift operator, and it is regular when its first and last coefficients…

Let an equation possess a strong symmetry operator, and suppose that a Bäcklund transformation leads to the relation … where $u$ is a solution and $\sigma_1$ is an eigenfunction as…

Let a potential and its eigenfunctions satisfy shape invariance, meaning that the corresponding Darboux-transformed eigenfunctions and potentials occur in pairwise shape-invariant…

Darboux-reducibility conjecture. $W$ is not a Darboux transformation of a direct sum of classical weights, but it is Darboux-reducible.

Nonexistence conjecture.

Let $p$ be the sequence of orthonormal polynomials satisfying $xp=J_{N+1}p$, where $J_{N+1}$ is the Jacobi matrix associated with the measure $(x-c)^{N+1}d\mu(x)$. Suppose that the…

Let $T(q)$ be the CKP Darboux transformation associated with a solution $q$, let $g\in Sp_{\infty}$, and write the corresponding tau function as … Let…

dQM deformation conjecture. For an appropriate range of the parameters, the deformed Hamiltonian $\mathcal{H}_{\mathcal{D}}$ is well-defined and hermitian.

Let an $A_{2k}$-Painlevé system be the higher-order Painlevé system associated with the cyclic dressing-chain construction discussed in the source, and let the construction describ…

A rational odd cyclic dressing chain is a cyclic chain of Darboux transformations with rational data and odd length, and an associated rational solution of the $A_{2n}$-Painlevé sy…

An exceptional polynomial system is a polynomial family associated with an exceptional differential operator, while a classical polynomial system is one of the classical orthogonal…

Let $P_n$ be a sequence of monic polynomials obtained as eigenpolynomials of a linear differential operator solving the classical Bochner–Krall problem. Let $T$ and $T^{-1}$ denote…

Gómez-Ullate–Kamran–Milson conjecture. Every exceptional orthogonal polynomial system of codimension $m$ can be obtained by applying a sequence of at most $m$ Darboux transformatio…

Let a two-dimensional Schrödinger-type operator be an operator of the form $\partial_x\partial_y+a\partial_x+b\partial_y+c$, and let a Darboux transformation be a transformation be…

m-step Darboux conjecture. Every $\mathrm{X}_m$ orthogonal polynomial system for any codimension $m$ can be obtained by applying a sequence of at most $m$ Darboux transformations t…

Darboux-Crum conjecture. Every exceptional orthogonal polynomial system is related to a classical orthogonal polynomial system by a Darboux-Crum transformation.

Let $\hat{H}_{SG}$ be the sine-Gordon Hamiltonian, let $\hat{\Upsilon}$ be any intertwiner identified above, and let $\lambda$ be an eigenvalue of $\hat{H}_{SG}$. A Darboux operato…

Let $N$ denote the dimension of the algebra used in the deformation-theoretic construction, and let the WDVV equation be the nonlinear PDE system arising from the reduction $C=1$,…

Let $P_I(x)$ be the polynomials associated with the Darboux-transformed potentials $V_I$ in the construction above. The potentials $V_I$ are obtained from these polynomials through…