6 problems

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 $d\geq2$, $q\in(\frac{d+1}{2},d)$, $p>\frac{(d-1)q}{d-q}$, and let $V\in L^q(\mathbb{R}^d)$ be radial. If $z_j$ are the eigenvalues of $H=-\Delta+V$, write $\delta(z_j)$ for th…

Isochrony in radial potentials is described through Keplerian interpretations of isochrone orbits, parabolic representations in Hénon's variables, and linear transformations shaped…

Consider the eikonal problem for radial potentials, and let $f(x,y;k)$ denote its solution. In the explicitly solvable Lorentzian case, singularities appear at certain points of th…