10 problems

Let $E_{\ell}^{(0)}$ denote the ground-state energy associated with the non-negative integer angular-momentum value $\ell$. Grosse–Martin conjecture. Knowledge of the ground-state…

Let $q\in L^2(0,1)$ be a radial potential, and for each nonnegative integer $\ell$ let the associated Dirichlet spectrum be the spectral data for angular momentum $\ell$. Two poten…

Finite-kernel conjecture. The differential $d_0\mathcal{S}_{\ell_1,\ell_2}$ has a finite-dimensional kernel for every pair of distinct nonnegative integers $\ell_1\neq\ell_2$. The…

Let $q\in L^2(0,1)$ be a radial potential, and for each nonnegative integer angular momentum $\ell$ let $\{\lambda_{\ell,n}(q)\}_{n\geq 1}$ denote the Dirichlet spectrum of the cor…

Let $T_q$ and $T_{\tilde{q}}$ be Sturm–Liouville operators with potentials $q$ and $\tilde{q}$ on $[0,\pi]$. Suppose that they have the same Weyl class. Weyl-class uniqueness conje…

The spectral datum of a Hankel operator consists of the type of its spectral measure and its multiplicity function. For compact Hankel operators, Gérard and Grellier constructed a…

Let $\eta=\{\eta_k\}_{k\geq1}$ and $\eta^{(1)}=\{\eta_k^{(1)}\}_{k\geq1}$ be the frequency collections associated with the comb domains describing the KdV flows. Comb-domain generi…

Let $\mathsf{E}$ be the spectral set, let $\mathcal{R}(\mathsf{E})$ denote the reflectionless potentials with spectrum $\mathsf{E}$, and let … be the divisor map. Let…

Higher-dimensional Borg conjecture. The following claims are equivalent: 1. The potential $V$ is constant. 2. There exists an entire function whose graph $\lambda=f(k)$ belongs to…

Let $\Gamma$ be the Riemann surface associated with the periodic operator, and let $\phi(x;\lambda)$ be its normalized Floquet solution. If $\Gamma$ has finitely many ramification…