Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution

TL;DR

This paper establishes sharp bounds for spectral projectors on surfaces of revolution with narrow bandwidths, extending previous results to smaller bandwidths using microlocal analysis and quantum integrability.

Contribution

It introduces new bounds for spectral projectors with polynomially narrow bandwidths on surfaces of revolution, utilizing microlocal techniques and quantum integrability structures.

Findings

Bound of order λ^{1/2} δ^{1/2} for spectral projector norms

Applicable for δ ≥ λ^{-1/32} on a broad class of surfaces of revolution

First sharp results for δ << 1 beyond symmetric surfaces

Abstract

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).