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

Abstract

It has been known since the work of Avakumovíc, Hörmander and Levitan that, on any compact smooth Riemannian manifold, if $-\hat{I}{”}_g\ddot{I}{ˆ}_{\hat{I}}»=\hat{I}»\ddot{I}{ˆ}_{\hat{I}}»$, then $\Vert \ddot{I}{ˆ}_{\hat{I}}»{\Vert }_{L^{\infty }}\le C\hat{I}{»}^{\frac{d-1}{4}}\Vert \ddot{I}{ˆ}_{\hat{I}}»{\Vert }_{L^2}$. It is believed that, on manifolds of negative curvature, such a bound can be largely improved; however, only logarithmic improvements in $\hat{I}»$ 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.

25 pages, minor corrections