Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution

Given a compact surface of revolution with Laplace-beltrami operator $\mathrm{\Delta }$, 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 $\sqrt{-\mathrm{\Delta }}$. For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the $L^2\to L^{\mathrm{\infty }}$ norm of $P_{\lambda ,\delta }$ is of order ${\lambda }^{\frac{1}{2}}{\delta }^{\frac{1}{2}}$ up to $\delta \ge {\lambda }^{-\frac{1}{32}}$. 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 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 $\delta \ll 1$ beyond the case of locally symmetric surfaces (torus, sphere, arithmetic hyperbolic surfaces).