Effective Dynamics of Translationally Invariant Magnetic Schr\"odinger Equations in the High Field Limit

TL;DR

This paper investigates the behavior of magnetic Schr"odinger equations in high magnetic field regimes, deriving an approximate solution description and demonstrating its stability under certain perturbations.

Contribution

It introduces a novel asymptotic analysis of magnetic Schr"odinger equations with translational invariance, including stability results using almost invariant subspaces.

Findings

Derived an approximate solution description using perturbation theory.

Identified high-frequency oscillations and magnetic drifts in the effective dynamics.

Proved stability of the asymptotic description under specific perturbations.

Abstract

We study the large field limit in Schr\"odinger equations with magnetic vector potentials describing translationally invariant $B$-fields with respect to the $z$-axis. In a first step, using regular perturbation theory, we derive an approximate description of the solution, provided the initial data is compactly supported in the Fourier-variable dual to $z\in \mathbb R$. The effective dynamics is thereby seen to produce high-frequency oscillations and large magnetic drifts. In a second step we show, by using the theory of almost invariant subspaces, that this asymptotic description is stable under polynomially bounded perturbations that vanish in the vicinity of the origin.