On semi-classical spectral series for an atom in a periodic polarized electric field

Abstract

In this report we present preliminary results about the tunneling problem for a magnetic Schrödinger operator. As a motivation we consider the 3-D time-dependent Schrödinger operator $H(t)=-h^2\Delta +V+E(t)\cdot x$ where $V$ is a radial potential and $E(t)$ a circularly polarized field with uniform frequency $\omega$. The quantum monodromy operator (QMO) that takes the system through a complete period $T=2\pi /\omega$, turns out to be unitarily equivalent to $e^{iT{P}_A(x,h{D}_x)/h}$, where $P_A(x,h{D}_x))$ identifies with a magnetic Schrödinger operator. When $V$ is sufficiently confining, $P_A(x,h{D}_x))$ presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.