Related papers: Weyl Laws for Open Quantum Maps

The well known Weyl's Law (Weyl's asymptotic formula) gives an approximation to the number $\mathcal{N}_{\omega}$ of eigenvalues (counted with multiplicities) on a large interval $[0,\>\omega]$ of the Laplace-Beltrami operator on a compact…

We use a `trivial' delta method to prove the Weyl bound in $t$-aspect for the $\rm L$-function of a holomorphic or a Hecke-Maass cusp form of arbitrary level and nebentypus. In particular, this extends the results of Meurman and Jutila for…

In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author obtained a complex Bohr-Sommerfeld…

We study the counting function of Steklov eigenvalues on compact manifolds with boundary and obtain its upper bound involving the leading term of Weyl's law. Our estimate can be viewed as a weakened version of P\'{o}lya's Conjecture in the…

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph…

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting…

This article provides an Omega-result for the remainder term in Weyl's law for the spectral counting function of certain (2l+1)-dimensional Heisenberg manifolds.

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be…

We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…

We prove asymptotically optimal upper bounds for the eigenvalues of the Wentzel-Laplace operator on Riemannian manifolds with Ricci curvature bounded below. These bounds depend highly on the geometry of the boundary in addition to the…

We generalize the work of Lindenstrauss and Venkatesh establishing Weyl's Law for cusp forms from the spherical spectrum to arbitrary Archimedean type. Weyl's law for the spherical spectrum gives an asymptotic formula for the number of cusp…

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of $h$ which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with…

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend…

We derive explicit bounds for the remainder term in the local Weyl law for locally hyperbolic manifolds, we also give the estimates of the derivative of this remainder. We use these to obtain explicit bounds for the C^k-norms of the…

In this work we consider semi-classical Schr\"odinger operators with potentials supported in a bounded strictly convex subset ${\cal O}$ of ${\bf R}^n$ with smooth boundary. Letting $h$ denote the semi-classical parameter, we consider…

For bound states of atoms and molecules of $N$ electrons we consider the corresponding $K$-particle reduced density matrices, $\Gamma^{(K)}$, for $1 \le K \le N-1$. Previously, eigenvalue bounds were obtained in the case of $K=1$ and…

A method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new…

Let $b(k,\ell,\theta)$ be the maximum number of vertices of valency $k$ in a $(k,\ell)$-semiregular bipartite graph with second largest eigenvalue $\theta$. We obtain an upper bound for $b(k,\ell,\theta)$ for $0 < \theta < \sqrt{k-1} +…

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds…

Let $M$ be a smooth compact manifold of dimension $d$ without boundary. We introduce the concept of predominance for Riemannian metrics on $M$, a notion analogous to full Lebesgue measure which, in particular, implies density. We show that…