42 problems

Let $d\geq 2$, let $V:{\mathbb R}^d\to{\mathbb R}$ be periodic, and consider the continuous Schrödinger operator $-\Delta+V$. Bethe–Sommerfeld conjecture. The spectrum of…

Generic spectral-edge conjecture. For a generic such operator, each spectral-gap endpoint is a unique, modulo $2\pi\mathbb{Z}^d$, nondegenerate extremum of a single band function…

Ao's conjecture. The spectrum is pure point for $\alpha<0$, at least for “non-resonant” $F$, and is continuous for $\alpha>0$.

Spectral edges conjecture. Each extremum at a spectral edge should be attained at a unique point on the Bloch variety.

Recall that the spectrum of the operator $-\partial^2/\partial \mathbf{x}^2+Q(\mathbf{x})$, with $\mathbf{x}\in\mathbb{R}^3$ and $Q$ a real periodic potential, is absolutely contin…

Let $L$ be the periodic Schrödinger operator in equation (1), and let $\sigma(L)$ denote its spectrum. A solution is called well-decaying at infinity if it has the decay property i…

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…

Gieseker's irreducibility conjecture. The Floquet surface of $A$ is irreducible for every value of the spectral parameter $lambda$.

Complete Hessian degeneration conjecture. For some values of $t$, the bottom of the spectrum of $M_t$ exhibits complete degeneration, in the sense that its Hessian vanishes complet…

For a discrete periodic operator, let the dispersion relation be the band functions describing its spectrum as a function of quasimomentum. The extrema of these band functions are…

Let $H$ be a periodic elliptic second-order operator with sufficiently smooth coefficients. Kuchment's conjecture. The operator $H$ does not admit any flat bands. This conjecture i…

Let $Lu=\operatorname{div}(A\nabla u)+Vu$ be a second-order elliptic differential operator on $\mathbb{R}^d$, with real, bounded, $C^1$-smooth (or $C^{\infty}$-smooth) coefficients…

Guo's conjecture. The existence of a three-fold Weyl point extends from small potentials to large potentials.

Let $q(x)$ be an entire and $2\pi$-periodic function of $x$, and let … be the corresponding Hill operator. Real-interval spectrum conjecture. If the spectrum of $H$ is a real inter…

Let $\mathcal{L}$ be a periodic elliptic operator, and let its point spectrum mean the set of eigenvalues admitting corresponding square-integrable eigenfunctions. Absolute continu…

Let $\Omega$ be the period-$1/2$ waveguide structure, and let $-\Delta$ denote its operator. Suppose the spectral properties listed in Assumption 0 hold, including analyticity and…

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…

The periodic tubular structure is described by a magnetic Laplacian whose homogenised dispersion relation depends on the perturbation parameter, and for larger positive values of t…

Let $L$ be a sufficiently general discrete periodic operator, with band functions denoted by $\lambda_j$ on its Bloch variety. For generic parameters, an extreme value of the band…

Effective mass conjecture. For "generic" potentials, all band functions behave like Morse functions around their global minima and maxima.

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 $V$ be a non-constant real periodic potential on $\mathbb{Z}^2$, and let $F_{\lambda}(V)$ denote its Fermi variety at energy $\lambda\in\mathbb{C}$. Gieseker–Knörrer–Trubowitz…

Let $d\geq 2$, let $V$ be a periodic potential on $\mathbb{Z}^d$, and let $F_{\lambda}(V)$ denote the Fermi variety at energy $\lambda\in\mathbb{C}$. Fermi variety irreducibility 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…