Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution

Abstract

Given a compact surface of revolution with Laplace-beltrami operator $Δ$, we consider the spectral projector $P_{λ,δ}$ on a polynomially narrow frequency interval $[λ-δ,λ+ δ]$, which is associated to the self-adjoint operator $\sqrt{-Δ}$. 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 $P_{λ,δ}$ is of order $λ^{\frac{1}{2}} δ^{\frac{1}{2}}$ up to $δ\geq λ^{-\frac{1}{32}}$. We adapt the microlocal approach introduced by Sogge for the case $δ= 1$, by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdière. 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 $δ\ll 1$ beyond the case of locally symmetric surfaces (torus, sphere, arithmetic hyperbolic surfaces).