On the {$L^\infty $} norms of spectral projectors on shrinking intervals: the cases of some spheres of revolution and of the Euclidean disk
Ambre Chabert, Yves Colin de Verdìère
TL;DR
This work analyzes $L^2\to L^ fty$ bounds for spectral projectors on shrinking windows $[\lambda-\lambda^{-1/3},\lambda+\lambda^{ -1/3}]$ on compact 2D manifolds with quantum integrable geodesic flows. By exploiting action-angle coordinates, Lagrangian-oscillatory joint eigenfunctions, fold-type caustics, and a deformed lattice structure of the joint spectrum, the authors reduce the problem to counting estimates via Airy/BKW decay. They prove a polynomial improvement $\|P_{\lambda,\lambda^{-1/3}}\|_{L^2\to L^ fty(K)} \lesssim_K \lambda^{1/2-1/12}$ away from poles/equator on simple spheres of revolution and away from the disk center, with a refined exponent $1/18$ near the disk boundary. These results imply improved eigenfunction sup-norm bounds on compact sets and advance understanding of spectral-projection behavior in completely integrable geometric settings.
Abstract
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.