17 problems

Fix a finite spectral interval $\Lambda=(a,b)$. A periodic Hamiltonian $H$ is called generic on $\Lambda$ if, for every band edge $\lambda_0$ occurring inside $\Lambda$, the band f…

Consider a planar embedding of a periodic graph with equal angles between the tangent lines of the edges meeting at each vertex, and let $\Omega_\varepsilon$ be the associated fatt…

Generic gap-edge conjecture. Each endpoint of the gap is a unique, modulo the dual lattice, nondegenerate extremum of a single band function $\lambda_j(k)$.

Let $L=-\Delta+V(x)$ be a periodic Schrödinger operator, or more generally a second-order elliptic periodic operator, with Bloch variety $\mathcal{B}_L$. Bloch-variety irreducibili…

Consider an elliptic periodic operator and vary its parameters, for example its periodic potential. Generic gap-edge non-degeneracy conjecture. For almost all such parameters, the…

Let $L$ be a periodic second-order elliptic operator with sufficiently nice coefficients. Absolute continuity conjecture. The spectrum of $L$ is absolutely continuous, and hence…

Let $d\geq 3$, let $V_c:\mathbb{T}^d\to\mathbb{C}^d$ be a periodic function, and let $-\Delta_c+V_c$ be the corresponding continuous periodic Schrödinger operator, where $\Delta_c$…

Let $\mathcal B^D$ be the Bloch–Torrey operator on the periodically perforated domain, and let $\rho(\mathcal B^D)$ denote its resolvent set. Dense resolvent conjecture. … The pape…

Let $V$ be a real-valued periodic potential, let $H_0=-\Delta+V$ be the associated self-adjoint operator on $\ell^2(\mathbb{Z}^d)$, and let $\lambda_m(k)$, $k\in\mathbb{R}^d$, be i…

Let $d\geq 2$, let $V$ be a periodic potential, and let $F_{\lambda}(V)$ denote the Fermi variety at energy $\lambda$. An analytic set is irreducible if it cannot be represented as…

Let $V$ be a periodic potential and let $B(V)$ denote its Bloch variety. An analytic set is irreducible if it cannot be represented as the union of two nonempty proper analytic sub…

Let $V\in C^\infty(\mathbb{R}^3/\mathbb{Z}^3)$ be a smooth periodic potential, and consider the periodic Schrödinger operator $-\Delta_{\mathbb{R}^3}+V$. A degeneracy of its Bloch…

Consider a family of periodic operators whose parameters include the periodic potential, and possibly additional parameters such as a metric. Generic non-degeneracy means that dege…

Let $-elta+V$ be a Schrödinger operator on $\mathbb{R}^n$ with periodic potential, and let $\lambda_j(k)$ denote its band functions, with $k$ the quasimomentum. Spectral edges are…

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…

Fermi-surface irreducibility conjecture. For $\lambda\in\mathbb{R}$, the Fermi surface $F_{\lambda,H}$ is irreducible modulo $\Gamma^*$, possibly except for a discrete set of value…

Generic spectral-edge conjecture. Generically, the extrema of band functions are attained by a single band, are isolated, and are non-degenerate, meaning that their Hessians are no…