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

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. Using analytic 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 that this asymptotic description is stable under a fairly general class of singular perturbations by using the theory of almost invariant subspaces. To this end, we also give an elementary proof of the exponential decay of the resolvent and the eigenfunctions of confining Schr\"odinger operators.
Comment: 20 pages