14 problems

Let $\beta\in\mathbb C$ and $n\in\mathbb N$. Let $d_n^{(\beta)}$ denote the dimension or counting quantity defined in the source for the degree-$n$ states at momentum $\beta$, and…

Let $\mathscr{P}$ denote the set of partitions, let $E_k(\,\cdot\,;\epsilon,0)$ be the quantum KdV expression defined in the paper, and let $Q_j$ and $\ell(\nu)$ denote the shifted…

Let $\Lambda^*=\mathbb{Q}[Q_2,Q_3,\dots]$ be the algebra of shifted symmetric functions on partitions, and let $E_k^{[m]}(\lambda;c)$ be the expressions defined in the paper. Zagie…

Fix $d_0$ and $k>-2$, and let $r$ and the Gaudin and BLZ systems be as in the source. Define $Q_{+,r}(\lambda)=\lambda^rQ_+(\lambda)$ for the Gaudin spectral determinants, and use…

Fix $d\geq1$, $k>-2$, and $l>-\frac12$, and let $\bar k,\bar l$ be related to $k,l$ as in the source. Let the BLZ trivial-monodromy system and its spectral determinant…

Let $L^G$ be a Schrödinger operator obtained from a solution of the $chat{\mathfrak{sl}}_2$ Gaudin Bethe Ansatz equations, and let $Q_\pm(\lambda)$ be its connection coefficients,…

For each $k\geq 1$, let $\Sigma_{k,m}$, for $m\geq 1$, be the weight-$k$ spectral curve … Here $\mathscr{P}_k$ denotes the set of partitions of weight $k$, and $\ell(\lambda)$ deno…

Weak BLZ conjecture. For every $\alpha$ and $L$, the number of solutions of the BLZ system is at most $p(N)N!$. For generic values of $\alpha$ and $L$, the number of solutions is…

Let $P_n(N)$ be the number of $n$-coloured partitions of $N$, let $n=\operatorname{rank}\mathfrak{g}$, and let $\operatorname{systemonc}$ denote the displayed system of equations f…

Let a quantum $\widehat{\mathfrak{g}}$-KdV oper be an oper of the form … with the additional regular singularities chosen to have trivial monodromy for every value of $\lambda$. Le…

Affine Gaudin Bethe ansatz conjecture. Every eigenvalue of $\boldsymbol G_n$ in the level-$N$ subspace has the form $r_n(\bar s,\bar t)$ for a solution of the affine Gaudin Bethe a…

Litvinov's ILW Bethe ansatz conjecture. The spectrum of the ILW Hamiltonians is described by solutions of a distinct Bethe ansatz equation.

Litvinov's deformation conjecture. The algebra of local Hamiltonians can be naturally deformed by adding an extra Heisenberg algebra.

Gaudin Bethe ansatz conjecture. The spectrum of the non-local integrals of motion is described by solutions of Gaudin Bethe ansatz equations associated to affine…