# On large potential perturbations of the Schr\"odinger, wave and Klein--Gordon equations

## TL;DR

This paper establishes sharp resolvent estimates for magnetic Schr"odinger operators with large potentials, leading to improved smoothing and Strichartz estimates for related quantum and wave equations.

## Contribution

It provides the first sharp resolvent bounds for large, almost critically decaying potentials in magnetic Schr"odinger operators, enabling enhanced dispersive estimates.

## Key findings

- Proved sharp resolvent estimates for magnetic Schr"odinger operators.
- Derived improved smoothing estimates for Schr"odinger, wave, and Klein-Gordon equations.
- Established Strichartz estimates under large potential conditions.

## Abstract

We prove a sharp resolvent estimate in scale invariant norms of Amgon--H\"{o}rmander type for a magnetic Schr\"{o}dinger operator on $\mathbb{R}^{n}$, $n\ge3$\begin{equation*} L=-(\partial+iA)^{2}+V \end{equation*}with large potentials $A,V$ of almost critical decay and regularity. The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schr\"{o}dinger, wave and Klein--Gordon flows associated to $L$.