28 problems

Let $\Omega\subset\mathbb{R}^d$ be a domain with Dirichlet Laplacian eigenvalues $\{\lambda_k\}$, let $|\Omega|$ denote its volume, and let $\omega_d$ be the volume of the unit bal…

Let $M:=M_1\times\ldots\times M_n$ be the product of at least two compact Riemannian manifolds without boundary, and let $d$ be its dimension. Denote by $N_M(\lambda)$ the eigenval…

Let $Omega subset R^n$ be a sufficiently regular domain, and let $ND( Omega,t)$ denote its Dirichlet Laplace eigenvalue counting function. Write…

Let $R(\lambda)$ denote the remainder in the integrated Weyl law for the eigenvalue counting function on a surface of constant negative curvature. Randol's conjecture. For every…

Let $\textbackslash Sigma\textbackslash subset\textbackslash mathbb R^d$ be a domain with $d\textbackslash ge1$, and let $N(\textbackslash lambda)$ count the Dirichlet eigenvalues…

Let $\mathscr{D}\subset\mathbb{R}^d$ be a bounded Euclidean domain with $d\geq 2$, and consider the Dirichlet or Neumann Laplacian on $\mathscr{D}$. If…

Let $Omega$ be a domain with boundary $partial Omega$, and let $ND( Omega,t)$ denote its Dirichlet Laplace eigenvalue counting function. When $partial Omega$ is irregular, its H…

For a compact surface with a non-arithmetic or generic geodesic flow, let $R_h$ denote the remainder in the Weyl law for the semiclassical Laplacian. The $O(h^{2-})$ remainder conj…

Let $G$ be a real reductive group, $K_{\infty}$ a maximal compact subgroup, and let $d=\operatorname{dim}(G(\mathbb{R})/K_{\infty})$. Let $\Gamma$ be an arithmetic subgroup and let…

Let $S^{d_1}\times S^{d_2}\times\cdots\times S^{d_n}$ be a product of spheres of length $n\geq 5$, let $d=d_1+\cdots+d_n$, and suppose the Weyl error bounds have the form…

Let $\Omega\subset\mathbb{R}^d$ be a Euclidean domain with piecewise smooth boundary, and let $\mathcal{N}_\Omega(\lambda)$ be its eigenvalue counting function. The set of periodic…

Let $\mathcal{T}$ be a tetrahedron, and let $\mu_k$ denote the eigenvalues of the homogeneous Dirichlet Laplacian on $\mathcal{T}$, ordered so that … Write $|\mathcal{T}|$ and…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, let $N(\lambda)$ be its eigenvalue counting function, and write the Weyl law as … Here $E(\lambda)$ is the Weyl remai…

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$, let $\Delta_g$ be the negative definite Laplace–Beltrami operator, and let $\{\lambda_j^2\}_{j=0}^\infty$ be t…

Weyl's perimeter conjecture. As $\mu\to\infty$,

Weyl–Schanuel law. As $Q\rightarrow\infty$,

Let $\Gamma$ be the boundary of a strictly convex obstacle, let $\gamma(y)$ be the dissipation parameter on $\Gamma$, and suppose … Let $N(r)$ denote the counting function of negat…

Let $D\subset\mathbb R^2$ be a bounded planar domain with piecewise smooth boundary. A billiard trajectory in $D$ is called periodic if it returns to its initial position and direc…

Berry's conjecture. The second term in the Weyl asymptotic should be replaced by

Hardy's conjecture. For $n=2$, the sharp exponent has the form

Let $\Omega$ be an arbitrary domain with discrete spectrum. A spectral core is the energy-dependent geometric subdomain introduced in the paper by retaining the sufficiently thick…

Let $M$ be a compact Riemannian manifold with boundary, and let $R(\lambda)$ denote the remainder term in the Weyl law for the eigenvalue counting function. Assume that the periodi…

Let $\Omega$ be an ellipse, and let $R(\lambda)$ denote the Weyl remainder in the spectral counting function for the Laplacian on $\Omega$. The paper establishes a lower bound for…

Let $\Omega\subset\mathbb{R}^n$ be a smooth bounded Euclidean domain, and let $G^t:S^*\Omega\to S^*\Omega$ be its billiard flow on the unit cotangent bundle. The non-periodicity co…

Let $X=\operatorname{SL}(2,\mathbb{Z})\backslash\mathcal{H}$ be the modular surface, and let $S(t)$ denote the Weyl remainder for its Maass spectrum. Square-root upper-bound conjec…