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Yves Colin de Verdìère

Publications and source records attributed to Yves Colin de Verdìère.

11 recordsLinked to original sources
arXiv2026

Weyl formulae for some singular metrics with application to acoustic modes in gas giants

This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace-Beltrami operator is presented and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.

math.AP↗
arXiv2026

On the {$L^\infty $} norms of spectral projectors on shrinking intervals: the cases of some spheres of revolution and of the Euclidean disk

Given a compact Riemannian surface $M$, with Laplace-Beltrami operator $Δ$, for $λ> 0$, let $P_{λ,λ^{-\frac{1}{3}}}$ be the spectral projector on the bandwidth $[λ-λ^{-\frac{1}{3}}, λ+ λ^{\frac{1}{3}}]$ associated to $\sqrt{-Δ}$. We prove a polynomial improvement on the $L^2 \to L^{\infty}$ norm of $P_{λ,λ^{-\frac{1}{3}}}$ for generic simple spheres of revolution (away from the poles and the equator) and for the Euclidean disk away from its center but up to the boundary. We use the Quantum Integrability of those surfaces to express the norm in terms of a joint basis of eigenfunctions for $\left(\sqrt{-Δ}, \frac{1}{i}\frac{\partial}{\partial θ}\right)$. Then, we use that those eigenfunctions are asymptotically Lagrangian oscillatory functions, each supported on a Lagrangian torus with fold-type caustic. Thus, studying the distribution of the caustics, and using BKW decay away from the caustics, we are able to reduce the problem to counting estimates.

math.AP↗
arXiv2024

Classical and Quantum mechanics on 3D contact manifolds

In this survey paper, I describe some aspects of the dynamics and the spectral theory of sub-Riemannian3D contact manifolds. We use Toeplitz quantization of the characteristic coneas introduced by Louis Boutet de Monvel and Victor Guillemin. We also discuss trace formulae following our work as well as the Duistermaat-Guillemin trace formula.

math.AP↗
arXiv2023

On essential-selfadjointness of differential operators on closed manifolds

The goal of this note is to present some arguments leading to the conjecture that a formally self-adjoint differential operator on a closed manifold is essentially self-adjoint if and only if the Hamiltonian flow of its symbol is complete. This holds for differential operators of degree two on the circle, for differential operators of degree one on any closed manifold and for generic Lorentzian Laplacians on surfaces.

math.AP↗
arXiv2022

Spectral asymptotics for sub-Riemannian Laplacians

We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.

math.DG↗
arXiv2022

Periodic geodesics for contact sub-Riemannian 3D manifolds

The goal of this paper is to study periodic geodesics for sub-Riemannian metrics on a contact 3D-manifold.We develop two rather independent subjects:1) The existence of closed geodesics spiraling around periodic Reeb orbits for a generic metric.2) The precise study of the periodic geodesics for a right invariant metric on a quotient of SL2(R)

math.DG↗
arXiv2019

Spectral theory of pseudo-differential operators of degree 0 and application to forced linear waves

We extend the results of our paper "Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0" written with Laure Saint-Raymond to the case of forced linear wave equations in any dimension. We prove that, in dimension 2,if the foliation on the boundary at infinity of the energy shell is Morse-Smale, we can apply Mourre's theory and hence get the asymptotics of the forced solution. We also characterize the wavefrontsets of the limit Schwartz distribution using radial propagation estimates.

math-ph↗
arXiv2018

Attractors for two dimensional waves with homogeneous Hamiltonians of degree 0

The density stratification in an incompressible fluid is responsible for the propagation of internal waves. In domains with topography, these waves exhibit interesting features. In particular, numerical and lab experiments show that, in two dimensions, for generic forcing frequencies, these waves concentrate on attractors. The goal of this paper is to analyze mathematically this behavior, using tools from spectral theory and microlocal analysis. The same results apply also to inertial waves in rotating fluids.

math-ph↗
arXiv2016

Topological Resonances on Quantum Graphs

In this paper, we try to put the results of Smilansky and al. on "Topological resonances" on a mathematical basis.A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exists compactly supported eigenfunctions. We give several estimateof the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case oftrees is also discussed.

math-ph↗