43 problems

Let $P u=-\Delta u+V(x)u+u$ be subcritical in $\mathbb{R}^n$, and assume that … If $u$ is a positive solution of $Pu=0$ in $\mathbb{R}^n$ satisfying the stated condition, then let…

Let $H_\varepsilon^\omega$ be the magnetic Schrödinger operator obtained by replacing the approximating graph by a network with fat edge width $\varepsilon$, replacing the $\delta$…

Let $\Gamma$ be the orbifold fundamental group of a compact good orbifold, acting on its universal orbifold cover $\widetilde{M}$, and let $H_V=\Delta+V$ be a $\Gamma$-invariant Sc…

Turbiner's second conjecture. If the symbol of $H$ engenders a Euclidean geometry, then the spectral equation $H\psi=E\psi$ can be solved by separation of variables. The paper stat…

A quasi-exactly-solvable or exactly-solvable problem in $\mathbb{R}^2$ is one containing the Laplace–Beltrami operator with a flat-space metric tensor; its variables are non-separa…

Let $M$ be a Riemannian manifold, let $E$ be a Hermitian vector bundle over $M$, and let $V$ be a potential satisfying Assumption A: $V=V_{+}+V_{-}$, with $V_{+}(x)8e0$ and…

Let $H_0=-\Delta+q(x)$ be a Schrödinger operator with periodic potential $q$, let $F_{\,\lambda}(q)$ be its Fermi variety, and let the Condition mean that, for every $\lambda$ in t…

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…

Let $H=-\Delta+V(x)$ act on $\mathbb{R}^{d}$, where $V$ is a potential satisfying … Here $\sigma_{ac}(H)$ denotes the absolutely continuous spectrum of $H$, and $\mathbb{R}^{+}$ de…

General-potential conditioning conjecture. As $T\to\infty$, the probability measures

Let $d\geq2$ and $q\in(d/2,d]$. Let $V$ be a bounded complex-valued potential on $\mathbb{R}^d$, and let $z$ be an eigenvalue of $-\Delta-V$ with…

Let $Y$ be a finite set of points on a loop $\Gamma$ of fixed length in $\mathbb{R}^{\nu}$, where $\nu=2,3$, with the points equidistant in the arc-length variable, and suppose tha…

Fundamental gap conjecture. The fundamental gap is at least $3\pi^2/D^2$.

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a domain, possibly equal to $\mathbb{R}^d$, and let $V:\Omega \to \mathbb{R}$ be a sufficiently smooth potential bounded from belo…

Let $\tau>0$ and let $\function{P}_\tau$ denote the spectral projector associated with $-\Delta+|x|^\tau$. Suppose that $\omega\five \subset\real^d$ is measurable and that, for some…

Let $D$ be the discrete Molchanov–Vainberg Laplacian in dimension $2$, let $\kappa$ be even, and let $\mathcal{E}_{i_1},\mathcal{E}_{i_2}\in\boldsymbol{\Theta}_{\kappa}(D)$ be cons…

For even $\kappa\geq6$, define $J_3(\kappa)=(\cos(2\pi/\kappa),\cos^2(\pi/\kappa))$, and let $\boldsymbol{\mu}_{\kappa}(D)$ denote the absolutely-continuous-spectrum set defined in…

Let $D$ be the discrete Molchanov–Vainberg Laplacian, let $\boldsymbol{\Theta}_{m,\kappa}(D)$ denote the threshold sets defined in the paper, and let $\mathbb{N}_e$ denote the even…

Let $d\geq 2$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be a periodic potential such that the Schrödinger operator $H_V$ is self-adjoint. Bethe–Sommerfeld conjecture. The spectrum…

In $\mathbb{R}^d$, where $d$ is an arbitrary positive integer, let $H=-\Delta+V$ be a Schrödinger operator with $V(\boldsymbol{x})$ of unbounded support. Suppose that $H$ has a bou…

Let $V$ be a single-well potential with centered transition point $\tau=0$ or a convex potential, and let $\Lambda(V,\alpha)$ denote the fundamental spectral gap of the one-dimensio…

Let $L(V)$ be the operator in the paper, and let periodic and antiperiodic eigenvalues mean Bloch eigenvalues with the corresponding periodic or antiperiodic boundary condition. Si…

Let $H(c)$ be the operator with pure imaginary potential $2ic\cos 2x$, and let $L(V)$ be the corresponding operator whose Bloch eigenvalues depend on $V$. Leaving principle. If $c$…

Weaker optimality conjecture. There exist $c_1,c_2,m_1,m_2>0$ such that

Optimality conjecture. There exist $c_1,c_2>0$ such that