Improved $L^\infty$ bounds for eigenfunctions under random perturbations in negative curvature

Abstract

It has been known since the work of Avakumov\'ic, H\"ormander and Levitan that, on any compact smooth Riemannian manifold, if $-\Delta_g \psi_\lambda = \lambda \psi_\lambda$, then $\|\psi_\lambda\|_{L^\infty} \leq C \lambda^{\frac{d-1}{4}} \|\psi_\lambda\|_{L^2}$. It is believed that, on manifolds of negative curvature, such a bound can be largely improved; however, only logarithmic improvements in $\lambda$ have been obtained so far. In the present paper, we obtain polynomial improvements over the previous bound in a generic setting, by adding a small random pseudodifferential perturbation to the Laplace-Beltrami operator.