Related papers: On the remainder in the Weyl formula for the Eucli…

In \cite{colin}, Y. Colin de Verdi\`ere proved that the remainder term in the two-term Weyl formula for the eigenvalue counting function for the Dirichlet Laplacian associated with the planar disk is of order $O(\lambda^{2/3})$. In this…

This article is concerned with estimations from below for the remainder term in Weyl's law for the spectral counting function of certain rational (2l+1)-dimensional Heisenberg manifolds. Concentrating on the case of odd l, it continues the…

In this paper, by explicitly calculating the principal symbols of pseudodifferential operators and by applying H\"omander's spectral function theorem, we obtain the Weyl-type asymptotic formulas with sharp remainder estimates for the…

The classical Weyl Law says that if $N_M(\lambda)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $\lambda$, then $$…

We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy.…

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.

Let $\Omega$ be a bounded domain with $C^\infty$ boundary in an $n$-dimensional $C^\infty$ Riemannian manifold, and let $\varrho$ be a non-negative bounded function defined on $\partial \Omega$. It is well-known that for the biharmonic…

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order $(1,1)$, on an asymptotically Euclidean manifold. We first prove a two term…

In this paper, we introduce a Weyl functional calculus $a \mapsto a(Q,P)$ for the position and momentum operators $Q$ and $P$ associated with the Ornstein-Uhlenbeck operator $ L = -\Delta + x\cdot \nabla$, and give a simple criterion for…

We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $(\lambda_i)_{i=1}^\infty$ be the variational spectrum of $\Delta_p$ on a closed Riemannian manifold $(X,g)$ and let $N(\lambda) = \#\{i:\,…

We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold $(M,g)$ of dimension $n$, let $\Pi_\lambda$ denote the kernel of the spectral projector…

Let X=Sl(3,Z)\Sl(3,R)/SO(3,R). Let N(lambda) denote the dimension of the space of cusp forms with Laplace eigenvalue less than lambda. We prove that N(lambda)=C lambda^(5/2)+O(lambda^2) where C is the appropriate constant establishing…

On a compact Riemannian manifold $M$ of dimension $n$, we consider $n$ eigenfunctions of the Laplace operator $\Delta $ with eigenvalue $\lambda$. If $M$ is homogeneous under a compact Lie group preserving the metric then we prove that the…

We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods.

Let $M= \Gamma \setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $\Gamma$. We show that the eigenvalue counting function $N(\lambda)$ for any fixed element of a family…

We compute the large N limit of the partition function of the Euclidean Yang--Mills measure with structure group SU(N) or U(N) on all closed compact surfaces, orientable or not, excepted for the sphere and the projective plane. This limit…

We study the spectral asymptotics of wave equations on certain compact spacetimes where some variant of the Weyl asymptotic law is valid. The simplest example is the spacetime $S^1 \times S^2$. For the Laplacian on $S^1 \times S^2$ the Weyl…

(1) Suppose $\mu$ is a smooth measure on a hypersurface of positive Gaussian curvature in $\R^{2n}$. If $n\ge 2$, then $W(\mu)$, the Weyl transform of $\mu$, is a compact operator, and if $p>n\ge 6$ then $W(\mu)$ belongs to the $p$-Schatten…

Topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition function has a…

In this work we consider the Neumann problem for the Laplace operator and we prove an existence result in the H\"older spaces and obtain Schauder estimates. According to our knowledge this result is not explicitly proved in the several…