arXiv subjects

Yannick Bonthonneau

Publications and source records attributed to Yannick Bonthonneau.

7 recordsLinked to original sources

arXiv2017

Weyl laws for manifolds with hyperbolic cusps

We give Weyl-type estimates on the natural spectral counting function for manifolds with exact hyperbolic cusps. We treat three different cases: without assumption on the compact part, assuming that periodic geodesics form a measure-zero set, and assuming the curvature is negative. In each case, we obtain the same type of remainder as in the corresponding case in the context of compact manifolds. We also investigate the counting of resonances. In particular, we extend results of Selberg to the case of non-constant, negative curvature metrics, under a genericity assumption.

math.SP↗

arXiv2017

WKB constructions in bidimensional magnetic wells

This article establishes, in an analytic framework and in two dimensions, the first WKB constructions describing the eigenfunctions of the pure magnetic Laplacian with low energy when the magnetic field has a unique minimum that is positive and non-degenerate.

math.SP↗

arXiv2016

A lower bound for the $Θ$ function on manifolds without conjugate points

In this short note, we prove that the usual $Θ$ function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. This elementary lemma implies that the Bérard remainder in the Weyl law is valid for a manifold without conjugate points, without any restriction on the dimension.

math.DG↗

arXiv2015

Quantum Ergodicity for Eisenstein functions

A new proof is given of Quantum Ergodicity for Eisenstein Series for cusped hyperbolic surfaces. This result is also extended to higher dimensional examples, with variable curvature.

math.AP↗

arXiv2015

Resonance-free regions for negatively curved manifolds with cusps

The Laplace-Beltrami operator on cusp manifolds has continuous spectrum. The resonances are complex numbers that replace the discrete spectrum of the compact case. They are the poles of a meromorphic function $φ(s)$, $s\in \mathbb{C}$, the \emph{scattering determinant}. We construct a semi-classical parametrix for this function in a half plane of $\mathbb{C}$ when the curvature of the manifold is negative. We deduce that for manifolds with one cusp, there is a zone without resonances at high frequency. This is true more generally for manifolds with several cusps and generic metrics. We also study some exceptional examples with almost explicit sequences of resonances away from the spectrum.

math.SP↗

arXiv2014

Long Time Quantum Evolution of Observables

We build a semi-classical quantization procedure for finite volume man- ifolds with hyperbolic cusps, adapted to a geometrical class of symbols. We prove an Egorov Lemma until Ehrenfest times on such manifolds. Then we give a version of Quantum Unique Ergodicity for the Eisenstein series for values of s converging slowly to the unitary axis.

math.SP↗

arXiv2014

A note on the resonance counting function for surfaces with cusps

We prove sharp upper bounds for the number of resonances in boxes of size 1 at high frequency for the Laplacian on finite volume surfaces with hyperbolic cusps. As a corollary, we obtain a Weyl asymptotic for the number of resonances in balls of size $T \to \infty$ with remainder $O(T^{3/2})$.

math.SP↗