Related papers: Quantitative version of Weyl's law
Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the…
The first purpose of this article is to provide conditions for a bounded operator in $L^2(\R^n)$ to be the Weyl (resp. anti-Wick) quantization of a bounded continuous symbol on $\R^{2n}$. Then, explicit formulas for the Weyl (resp.…
First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…
In this paper, we obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we…
When a Dirac semimetal is subject to a circularly polarized laser, it is predicted that the Dirac cone splits into two Weyl nodes and a nonequilibrium transient state called the Floquet Weyl semimetal is realized. We focus on the previously…
We obtain generalizations of classical versions of the Weyl formula involving Schr\"odinger operators $H_V=-\Delta_g+V(x)$ on compact boundaryless Riemannian manifolds with critically singular potentials $V$. In particular, we extend the…
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…
A local UV cutoff $\Lambda(x)$ transforming under Weyl rescalings allows to construct Weyl invariant kinetic terms for scalar fields including Wilsonian cutoff functions. First we consider scalar fields in curved space-time with local bare…
We investigate the spectral asymptotic behavior of operator-valued classical pseudo-differential operators ($\Psi$DOs) for negative order with symbols taking values in a semifinite von Neumann algebran $\mathcal{M}$ equipped with a normal…
We consider the remainder term in the semiclassical limit formula for the eta invariant on a metric contact manifold, proving in general that it is controlled by volumes of recurrence sets of the Reeb flow. This particularly gives a…
We prove the extensions of Birkhoff's and Cotlar's ergodic theorems to multi-dimensional polynomial subsets of prime numbers $\mathbb{P}^k$. We deduce them from $\ell^p$-boundedness of $r$-variational seminorms for the corresponding…
We derive a representation formula for the Weyl solution to the Schr\"odinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with…
We present a semiclassical calculation of the generalized form factor which characterizes the fluctuations of matrix elements of the quantum operators in the eigenbasis of the Hamiltonian of a chaotic system. Our approach is based on some…
The paper is concerned with the asymptotic distribution of Laplace eigenvalues on Liouville tori. Liouville metrics are the largest known class of integrable metrics on two-dimensional tori; they contain flat metrics and metrics of…
We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner…
Let $S$ be a spinor bundle of a pseudo-Euclidean vector bundle $(E,\mathrm{g})$ of even rank. We introduce a new filtration on the algebra $\mathcal{D}(M,S)$ of differential operators on $S$. As main property, the associated graded algebra…
The quantum Hamiltonian generates in time a family of evolution operators. Continuity of this family holds within any choice of representation and, in particular, for the Weyl propagator, even though its simplest semiclassical approximation…
In this work, we consider a class of second order uniformly elliptic operators with smooth and bounded coefficients. We provide some estimates on the norm of the semigroup generated by these operators acting on weighted Sobolev spaces,…
The concept of translation of an operator allows to consider the analogous of shift-invariant subspaces in the class of Hilbert-Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the…
We obtain sharp uniform bounds on the low lying eigenfunctions for a class of semiclassical pseudodifferential operators with double characteristics and complex valued symbols, under the assumption that the quadratic approximations along…