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

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) = −h²∆ + V + E(t) • x, where V is a radial potential and E(t) is a circularly polarized field with uniform frequency ω. The quantum monodromy operator (QMO) that takes the system through a complete period T = 2π/ω, turns out to be unitarily equivalent to exp{iTPA(x,hDx)/h}, where PA(x,hDx)) identifies with a magnetic Schrödinger operator. When V is sufficiently confining, PA(x,hDx)) presents a double magnetic well. Then we construct its semi-classical ground state and examine the splitting between its two first eigenvalues.