# Semi-classical resolvent estimates for l $\infty$ potentials on Riemannian manifolds

## TL;DR

This paper establishes semi-classical resolvent estimates for Schrödinger operators with bounded potentials on various non-compact Riemannian manifolds, revealing how the manifold's geometry influences the bounds.

## Contribution

It provides new resolvent estimates for Schrödinger operators with L∞ potentials on different types of non-compact manifolds, highlighting the impact of geometric structure at infinity.

## Key findings

- Resolvent bounds depend on manifold structure at infinity.
- For Euclidean-like manifolds, bounds are of the form exp(Ch^{-4/3} log(h^{-1})).
- For hyperbolic manifolds, bounds are of the form exp(Ch^{-4/3}).

## Abstract

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the man-ifold at infinity. In particular, we show that for compactly supported real-valued L $\infty$ potentials and asymptoticaly Euclidean manifolds the resolvent bound is of the form exp(Ch --4/3 log(h --1)), while for asymptoticaly hyperbolic manifolds it is of the form exp(Ch --4/3), where C > 0 is some constant.