# Resolvent estimates for the magnetic Schr\"odinger operator in dimension $n \geq 2$

## TL;DR

This paper extends known resolvent estimates for magnetic Schr"odinger operators from higher dimensions to all dimensions greater than or equal to two, demonstrating similar decay properties in weighted spaces.

## Contribution

It proves that high-frequency resolvent estimates for magnetic Schr"odinger operators hold in all dimensions n ≥ 2, broadening previous results limited to n ≥ 3.

## Key findings

- Resolvent norms decay like λ^{-1/2} at high energy.
- Estimates valid for large long or short range potentials.
- Results apply uniformly across all dimensions n ≥ 2.

## Abstract

It is well known that the resolvent of the free Schr\"odinger operator on weighted $L^2$ spaces has norm decaying like $\lambda^{-\frac{1}{2}}$ at energy $\lambda$. There are several works proving analogous high-frequency estimates for magnetic Schr\"odinger operators, with large long or short range potentials, in dimensions $n \geq 3$. We prove that the same estimates remain valid in all dimensions $n \geq 2$.