1,639 problems

Let $\zeta(s)=\sum_{n\ge1}n^{-s}$ be the Riemann zeta function, continued meromorphically to $\mathbb{C}$. Its nontrivial zeros are the zeros $\rho=\beta+i\gamma$ in the critical s…

Selberg's eigenvalue conjecture. For every $j$, $\theta_j=0$.

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain. Let $\mu_n(\Omega)$ be the Neumann eigenvalues of the Laplacian, enumerated with multiplicity, and define … With … wh…

Let $V(I')$ be the affine variety defined by the specialized ideal $I'$ associated with the specialized spectral-invariant system, and let $q$ denote the period. A point is regular…

Let $\phi_\lambda$ be an $L^2$-normalized Laplace eigenfunction on an arithmetic surface, with eigenvalue parameter $\lambda$. Iwaniec's conjecture. For every $\varepsilon>0$, one…

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…

The Schrödinger operator $H=-\Delta+V$ on $\mathbb{R}^{d}$ has negative eigenvalues $\lambda_n(-\Delta+V)$, and the Lieb–Thirring inequality is … Here $V_-(x)=\max\{0,-V(x)\}$, and…

Let $\mathbb{T}^d:=\mathbb{R}^d/(2\pi\mathbb{Z})^d$ be the $d$-dimensional torus, and let $\phi_\lambda$ be an $L^2$-normalized Laplace eigenfunction with eigenvalue parameter…

In a quantum mechanical system, let a semi-bounded operator be perturbed by a coupling constant, and regard the resulting partition function as a function of that coupling constant…

Weyl-law conjecture. A version of Weyl’s law should also apply to non-selfadjoint black-hole operators.

Laptev–Safronov conjecture. The estimate should remain valid throughout the larger range $q\le d$, rather than only for $\frac d2<q\le\frac12(d+1)$.

Jordan-curve conjecture. The following statements are equivalent:

Consider a model whose target spectral distribution is heavy-tailed, in contrast with the Marchenko–Pastur target distribution discussed above. Heavy-tailed target conjecture. A mo…

Jakobson–Naud's conjecture. They conjecture that

Aubry–André conjecture. The Lebesgue measure of the spectrum for any irrational frequency is

Let $A_{\Omega}[1]$ denote the operator associated with a domain $\Omega\subset\mathbb R^2$ and the constant symbol $1$, and let $\Omega$ be a convex domain. Flandrin's conjecture.…

Simon's conjecture. Cantor spectrum should be a generic phenomenon for one-dimensional almost-periodic operators.

Let $\Lambda_\varphi$ be the one-dimensional Dirac operator on $L^2({\mathbb{R}},{\mathbb{C}}^2)$, with potential $\varphi:{\mathbb{R}}\to{\mathbb{C}}$ satisfying … Call…

Petridis–Risager's exponential sum conjecture. For every $\epsilon>0$,

Let $\alpha\in{\mathbb R}\setminus{\mathbb Q}$, let $\Sigma_\alpha$ denote the spectrum of the critical almost Mathieu operator $H_{1,\alpha,\theta}$, and let $\dim_H$ denote Hausd…

Helffer–Nier's compact-resolvent conjecture. The operator $P$ is of compact resolvent if and only if its associated Witten Laplacian is of compact resolvent.

Let … be the Schrödinger operator with potential $u(x)=ix^3$. Bessis–Zinn-Justin conjecture. The spectrum of $H$ is real. This conjecture concerns the spectral reality of a non-sel…

Root-distribution conjecture. For each $k\geq -1$, the polynomial $P^{(\lambda)}_{l+4+k}$ is square free. Furthermore, for sufficiently large $k$, it has a fixed number of roots ou…

Let ${\mathcal H}$ be the $\mathfrak{U}_q(\mathfrak{gl}_n)$-invariant Hamiltonian, and let ${\mathfrak b}_{(N-k)k}$ be a state in the $k$-sector satisfying … For…

Let $H_\theta$ be the Harper operator with parameter $\theta$. Ten Martini conjecture. If $\theta$ is irrational, then the spectrum of $H_\theta$ is a Cantor set. The source states…