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

We consider the 3-D time-dependent Schrodinger operator H(t) = −h 2 ∆ + V + E(t) • x where V is a radial potential and E(t) a circularly polarized field with uniform frequency ω. Floquet operator that takes the system through a complete period T = 2π/ω, turns out to be unitarily equivalent to e iT P A (x,hDx)/h. Up to a linear gauge transformation, P A (x, hDx)) identifies with a magnetic Schrodinger operator with a double well. In case V is sufficiently confining, we construct the semi-classical ground state of P A and examine the splitting between its two first eigenvalues.