54 problems

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…

Non-closability conjecture. The sum $\mathcal{L}_{1,\mu}+\mathcal{L}_2$ is not closable.

BMS conjecture. Assume that $(M,g)$ is geodesically complete. Then $(M,g)$ is $L^{2}$-positivity preserving.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$, and let $m_{\frac{n-2}{4(n-1)}\operatorname{scal}_g}$ denote the mass associated with the conformal Laplacian. P…

Let $A$ be a self-adjoint elliptic operator of odd order on an even-dimensional manifold, and let its eta-invariant be defined in the usual sense. Gilkey's conjecture. The eta-inva…

Optimal-exponent conjecture. In each of these estimates, the power of $\varepsilon$ is optimal.

Let $H$ be the operator and $c_0$ the constant appearing in Theorem 4, whose boundary estimate has the form … for normalized eigenfunctions $Hf=\lambda f$. Optimality conjecture. T…

Let $U$ be a region, let $H$ be a uniformly elliptic second-order operator on $L^{2}(U)$ with Dirichlet boundary conditions, and let $Q$ be its associated quadratic form. For…

Let $n\ge 3$, let $\mathcal{E}$ be the class of all complex elliptic operators under study, and for $L\in\mathcal{E}$ let $p_-(L)$ be the lower endpoint of the maximal interval of…

Let $X$ be a compact space. A generalized elliptic operator over $X$ is a triple $(D,H_1,H_2)$, where $D:H_1\longrightarrow H_2$ is a Fredholm operator on Hilbert spaces $H_1$ and…

Let $M$ be a hyperkähler manifold that is AC of order $l\geq 1$, and let $k\geq -1$ be an integer. Let $x:M\rightarrow\mathbb{H}$ be a smooth function such that $D(x)$ is a $1$-for…

Let the first eigenfunction be normalized in $L_p$ for some $p>1$, and consider its associated limit measures in the setting of the paper. Normalization conjecture. Under $L_p$ nor…

Let $D$ be an elliptic differential operator. Its index is invariant under deformations of $D$, and the problem has associated topological data. Gelfand's conjecture. The index of…

Let $L_E$ be the Euclidean wave operator, and let the relation between the physical and fiducial spectral densities be denoted by … may remain valid, with some additional assumptio…

Let … be the distinguished family of proper biharmonic maps, and let $g(k)$ denote the number of mixed Fourier modes for which the corresponding explicit Fourier eigenvalue vanishe…

Equality conjecture. It is conjectured that

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain. Let $\lambda_k$ and $\mu_k$ denote, respectively, the Dirichlet and Neumann eigenvalues of the negative Laplacian on…

Let $L$ be the elliptic operator considered by Berestycki and Rossi, and let $lambda1'(L)$ and $lambda1”(L)$ denote their two generalised principal eigenvalues. Berestycki–Rossi'…

Expansion-pattern conjecture. The second-order expansions of

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 $K\subset\mathbb{R}^2$ be a Reifenberg flat set with sufficiently small flatness constant. A good operator is an operator of the type constructed in the paper, whose associated…

Let $u$ be a super-exponentially decaying solution of a second-order periodic, or more generally an appropriate class of periodic elliptic, equation. Landis-type conjecture. Such a…

Necessity conjecture. If $\partial\Omega$ is not uniformly rectifiable, then the displayed bound holds for none of the weak DKP operators.

Let $\Omega=(0,1)\times\widehat{\Omega}$, where $\widehat{\Omega}\subseteq\mathbb{R}^{d-1}$ is open and bounded, and let $\alpha\in L_\infty(0,1)$ be piecewise constant. For…

Let $\Omega\subset\mathbb{R}^n$ be a smooth exterior domain, and let $\Delta_D$ and $\Delta_N$ denote the Dirichlet and Neumann Laplacians on $\Omega$, respectively. The Hassell–Si…