Sharp resolvent estimates on non-positively curved asymptotically hyperbolic manifolds

TL;DR

This paper establishes optimal high-energy resolvent estimates for the Laplacian on non-positively curved asymptotically hyperbolic manifolds, improving previous bounds by analyzing the oscillatory nature of the kernel.

Contribution

It proves the sharp $O(| e \lambda|^{-1})$ resolvent bound under non-positive curvature, advancing understanding of spectral behavior on these manifolds.

Findings

Proves optimal resolvent bound $O(| e \lambda|^{-1})$ for non-positively curved AHM.

Analyzes oscillatory behavior of the resolvent's Schwartz kernel.

Improves upon previous polynomial bounds in the high-energy regime.

Abstract

We study the high energy estimate for the resolvent $R(\lambda)$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHM). In the literature, polynomial bound of the form $\|R(\lambda)\| = O(|\lambda|^{N})$ for $|\lambda|$ large and $\lambda\in \mathbb{C}$ in strips where $R(\lambda)$ is holomorphic was established for some $N > -1$. We prove the optimal bound $O(|\lambda|^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.