Two-body short-range systems in a time-periodic electric field (Q1847775)

Scattering problem for two \(\nu\)-dimensional particles interacting through a short-range potential and placed in an external time-periodic electric field may be described by the Hamiltonian \[ H(t)={p_1^2 \over {2m_1}} + {p_2^2 \over {2m_2}} -q_1 {\mathcal E}(t)\cdot x_1 -q_2 {\mathcal E}(t)\cdot x_2 + v(x_2-x_1)\;\text{ on} L^2({\mathbb R}^{2\nu}), \] where \(m_i\) and \(q_i, i=1,2\), are the masses and the charges of the two particles, respectively and \(x_1,x_2\) denote their positions. The electric field \(\mathcal E\) is periodic with some period \(T\), i.e. \({\mathcal E}(t+T) = {\mathcal E}(t)\) almost everywhere. The potential \(v\) is short-range with weak singularities, periodic in time with the same period as the field. Periodicity enables to compactify the time space variable. The resulting Hamiltonian \(\widehat H\) is called Floquet Hamiltonian. The author proves that the Floquet Hamiltonian has no bound states. Moreover, it is proved that wave operators exist and are unitary. Absence of a singular continuous spectrum for \(\widehat H\) is also proved.