Improved $L^\infty$ bounds for eigenfunctions under random perturbations in negative curvature
Maxime Ingremeau, Martin Vogel
TL;DR
This work addresses improved L^∞ bounds for Laplacian eigenfunctions on negatively curved manifolds by introducing small random pseudodifferential perturbations of the Laplacian. Using semiclassical analysis, noisy quantum dynamics, and a decomposition of eigenmodes into a finite superposition of unstable Lagrangian states, the authors prove high-probability polynomial improvements over the classical bound. The key contributions include a precise Lagrangian-state decomposition with controlled distortion, a propagation theory for superpositions under noisy dynamics, and a probabilistic argument yielding explicit exponent gains (Γ′) in the L^∞ norms, with concrete parameter choices in low dimensions. These results demonstrate that generic random perturbations enforce stronger delocalization of high-energy eigenfunctions, offering insights into quantum chaos in non-arithmetic settings and potential applications to spectral theory and PDEs on manifolds. The approach integrates classical dynamics (Anosov geodesic flow), microlocal analysis, and probabilistic techniques to achieve robust, high-probability improvements.
Abstract
It has been known since the work of Avakumovíc, Hörmander and Levitan that, on any compact smooth Riemannian manifold, if $-Δ_g ψ_λ= λψ_λ$, then $\|ψ_λ\|_{L^\infty} \leq C λ^{\frac{d-1}{4}} \|ψ_λ\|_{L^2}$. It is believed that, on manifolds of negative curvature, such a bound can be largely improved; however, only logarithmic improvements in $λ$ 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.