Long-time dynamics of coherent states in strong magnetic fields

Abstract

abstract:We consider the Schr\"odinger evolution of strongly localized wave packets under the magnetic Laplacian in the plane $\Bbb{R}^2$. When the initial energy is low, we obtain a precise control, in Schwartz seminorms, of the propagated states for times of order $1/\hbar$, where $\hbar$ is Planck's constant. In this semiclassical regime, we prove that the initial particle will always split into multiple coherent states, each one following the average dynamics of the guiding center motion but at its own speed, demonstrating a purely quantum ``ubiquity'' phenomenon.