arXiv subjects

Ambre Chabert

Publications and source records attributed to Ambre Chabert.

5 recordsLinked to original sources

arXiv2026

$L^4$ norm of spectral projectors on polynomially small frequency intervals for $S^1$-symmetric surfaces

For $(M,g)$ a compact Riemannian surface with Laplace-Beltrami operator $\Delta$, and for $\lambda,\delta \geq 0$, let $P_{\lambda,\delta}$ be the spectral projector on the frequency interval $[\lambda-\delta,\lambda+\delta]$ associated to $\sqrt{-\Delta}$. For the Euclidean disk, away from its boundary, we improve the upper bound on the $L^2\to L^4$ norm of $P_{\lambda,\delta}$ in the regime where the bandwidth $\delta$ is polynomially small compared to the target frequency $\lambda$. Decomposing on the explicit joint eigenbasis of $\left(\sqrt{-\Delta}, \frac{1}{i}\frac{\partial}{\partial \theta}\right)$ given in terms of Bessel eigenfunctions, which are well-approximated by oscillatory functions outside of their caustic set, we reduce the analysis to a number of precise quantitative estimates of nonstationary phase oscillatory integrals. We strongly use convexity phenomenon both for these estimates, and then for the summation of the contribution of all eigenfunctions through a new arithmetic estimate. The method extends to other $S^1$-symmetric surfaces satisfying similar conditions on the induced completely integrable structure.

math.AP↗

arXiv2026

Zonal states and improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces

We establish polynomially improved $L^\infty$ bounds for eigenfunctions of magnetic Laplacians on hyperbolic surfaces in the critical energy regime. We also show that, below the critical energy, the H\"ormander bound is saturated by explicit eigenstates, which we call magnetic zonal states. These states resemble zonal harmonics on the sphere and equidistribute on Lagrangian tori in phase space.

math.AP↗

arXiv2025

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 $\Delta$, for $\lambda > 0$, let $P_{\lambda,\lambda^{-\frac{1}{3}}}$ be the spectral projector on the bandwidth $[\lambda-\lambda^{-\frac{1}{3}}, \lambda + \lambda^{\frac{1}{3}}]$ associated to $\sqrt{-\Delta}$. We prove a polynomial improvement on the $L^2 \to L^{\infty}$ norm of $P_{\lambda,\lambda^{-\frac{1}{3}}}$ 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 $\left(\sqrt{-\Delta}, \frac{1}{i}\frac{\partial}{\partial \theta}\right)$. 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↗

arXiv2025

Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution

Given a compact surface of revolution with Laplace-beltrami operator $\Delta$, we consider the spectral projector $P_{\lambda,\delta}$ on a polynomially narrow frequency interval $[\lambda-\delta,\lambda + \delta]$, which is associated to the self-adjoint operator $\sqrt{-\Delta}$. For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the $L^2 \to L^{\infty}$ norm of $P_{\lambda,\delta}$ is of order $\lambda^{\frac{1}{2}} \delta^{\frac{1}{2}}$ up to $\delta \geq \lambda^{-\frac{1}{32}}$. We adapt the microlocal approach introduced by Sogge for the case $\delta = 1$, by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdi\`ere. This reduces the analysis to a number of estimates of explicit oscillatory integrals, for which we introduce new quantitative tools.This is the first sharp result in the case $\delta \ll 1$ beyond the case of locally symmetric surfaces (torus, sphere, arithmetic hyperbolic surfaces).

math.SP↗

arXiv2023

Weakly turbulent solution to Schr\"odinger equation on the two-dimensional torus with real potential decaying at infinity

We build a smooth time-dependent real potential on the two-dimensional torus, decaying as time tends to infinity in Sobolev norms along with all its time derivative, and we exhibit a smooth solution to the associated Schr\"odinger equation on the two-dimensional torus whose $H^s$ norms nevertheless grow logarithmically as time tends to infinity. We use Fourier decomposition in order to exhibit a discrete resonant system of interactions, which we are further able to reduce to a sequence of finite-dimensional linear systems along which the energy propagates to higher and higher frequencies. The constructions are very explicit and we can thus obtain lower bounds on the growth rate of the solution.

math.AP↗