Unitary Propagators for Time-Dependent Hamiltonians with Singular Potentials

AbstractWe prove that a unitary propagator U(t, s) for the time-dependent Schrödinger equation du/dt = iH(t)u exists in L2(R), where H(t) = −Δ + ∑Ni=1ciV(x − vit), ci, vi ∈ R, vi ≠ vj for i ≠ j, and V is a distribution with bounded Fourier transform. This extends earlier work with I. Segal on the time-independent case. Such Hamiltonians include, for example, the case of finitely many moving delta potentials. We also apply a method of Segal to study singular time-dependent perturbations of (1/i)(d/dx) in one space dimension and their corresponding unitary propagators, and show that the propagator depends continuously on the potential in a suitable sense