Semiclassical Resolvent Estimates for the Magnetic Schr{\"O}dinger Operator

TL;DR

This paper establishes semiclassical resolvent estimates for the magnetic Schr{"o}dinger operator in higher dimensions, providing bounds that depend on the regularity and decay properties of electric and magnetic potentials.

Contribution

It introduces new resolvent bounds for magnetic Schr{"o}dinger operators under general conditions, improving estimates for potentials with specific regularity and decay.

Findings

Bounded the weighted resolvent norm by exp(Ch^{-2} log(h^{-1}))

Achieved better bounds for H{"o}lder continuous potentials

Obtained resolvent bounds of exp(Ch^{-1}) for Lipschitz potentials

Abstract

We obtain semiclassical resolvent estimates for the Schr{\"o}dinger operator (ih$\nabla$ + b)^2 + V in R^d , d $\ge$ 3, where h is a semiclassical parameter, V and b are real-valued electric and magnetic potentials independent of h. Under quite general assumptions, we prove that the norm of the weighted resolvent is bounded by exp(Ch^{-2} log(h^{ -1} )) . We get better resolvent bounds for electric potentials which are H{\"o}lder with respect to the radial variable and magnetic potentials which are H{\"o}lder with respect to the space variable. For long-range electric potentials which are Lipschitz with respect to the radial variable and long-range magnetic potentials which are Lipschitz with respect to the space variable we obtain a resolvent bound of the form exp(Ch^{-1}) .