Longtime dynamics for the Landau Hamiltonian with a time dependent magnetic field

Abstract

We consider a modulated magnetic field, $B(t)=B_0+\epsilon f(\omega t)$, perpendicular to a fixed plane, where $B_0$ is constant, $\epsilon >0$ and $f$ a periodic function on the torus ${\mathbb{T}}^n$. Our aim is to study classical and quantum dynamics for the corresponding Landau Hamiltonian. It turns out that the results depend strongly on the chosen gauge. For the Landau gauge the position observable is unbounded for “almost all” non-resonant frequencies $\omega$. On the contrary, for the symmetric gauge we obtain that, for “almost all” non-resonant frequencies $\omega$, the Landau Hamiltonian is reducible to a two-dimensional harmonic oscillator and thus gives rise to bounded dynamics. The proofs use KAM algorithms for the classical dynamics. Quantum applications are given. In particular, the Floquet spectrum is absolutely continuous in the Landau gauge while it is discrete, of finite multiplicity, in symmetric gauge.