19 problems

Turbiner's second conjecture. If the symbol of $H$ engenders a Euclidean geometry, then the spectral equation $H\psi=E\psi$ can be solved by separation of variables. The paper stat…

Let $N$ be the number of particles, let $\psi_{\vec n}(y)$ be the one-variable factor appearing in the separated eigenfunction, and let $\vec n=(n_1,\ldots,n_N)$. Set $t=e^{2iy}$.…

Let $N$ be the number of particles, let $\boldsymbol q=(q_1,\ldots,q_N)$ be the coordinates, and let $\Psi_{\vec n}(\boldsymbol q)$ be an eigenfunction of the commuting Calogero–Su…

A quasi-exactly-solvable or exactly-solvable problem in $\mathbb{R}^2$ is one containing the Laplace–Beltrami operator with a flat-space metric tensor; its variables are non-separa…

Let $I_k$ be an integral adapted to Darboux–Haantjes coordinates $(Q_k,P_k)$, with parameters $\gamma_n$, and suppose that for $k=1,2$ its separated form is … while the Hamiltonian…

Three-parametric-solution conjecture. There are only two three-parametric, non-equivalent solutions of the equations … .

Self-annihilating invariance-vector-field conjecture. Among the invariance vector fields there exists a special vector field $Z$ which, acting in the system of natural Lie-algebrai…

A quadratically integrable system with integrals of nonstandard, nonseparable type may be related to a separable system only in a trivial manner. Non-subgroup-type integrals conjec…

Let $Z_{\kappa;\ell}$ be the partition function of the non-chiral WZW model with a network $\ell$ of Verlinde defects, and let $Z^W_{\kappa;\ell}$ be the corresponding partition fu…

Let $T_{1}^{(K)}(\lambda)$ be the transfer matrix of the general $gl_{1|2}$-graded Yang–Baxter algebra with twisted boundary conditions. Let $t_{1}(\lambda \mid \{x_a\})$ be the po…

Maillet–Nicolli conjecture. The separated variables coincide with the operatorial zeros of $\mathbb{B}(u)$.

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 an integrable quantum chain with transfer matrix and Sklyanin's $B$-operator, in the setting where the $B$-operator is diagonalizable and its eigenvalues have the stated f…

Consider the quantum chain and the operators, twists, inhomogeneities, and separation-of-variables bases appearing in Properties 1 and 2, where Property 1 concerns the diagonalizab…

In a spin chain with symmetry of rank higher than $\mathfrak{gl}_2$, an operator used to construct a separation-of-variables basis is considered together with the corresponding on-…

The SoV-operator conjecture. In addition to providing separated variables, the operator $B^{\rm good}(u)$ generates eigenstates of the transfer matrix, which have the form

The paper studies two-dimensional Hamiltonian systems admitting separation of variables in orthogonal coordinates and a third-order integral of motion. Subgroup-type coordinates pr…

Elliptic-coordinate nonexistence conjecture. In general elliptic coordinates, there are no non-trivial solutions of the linear compatibility conditions.

Let $H$ be a natural, nonintegrable Hamiltonian, and let $(H_i)$, $i=1,\ldots,n$, be a separable Stäckel system approximating its dynamics. Define $m_i=\{H,L_i\}=dL_i/dt$, where th…