Perturbation Theory for the Quantum Time-Evolution in Rotating Potentials

The quantum mechanical time-evolution is studied for a particle under the influence of an explicitly time-dependent rotating potential. We discuss the existence of the propagator and we show that in the limit of rapid rotation it converges strongly to the solution operator of the Schrödinger equation with the averaged rotational invariant potential. 1. The model, rotating frames We consider the dynamics of a quantum mechanical particle of mass m moving in R ; 2, with kinetic energy H0 = H0(p) = h(jpj) under the influence of a “rotating” potential V!t(x) = V0(R(!t) 1 x). One may think of an atom or molecule interacting, e.g., with the blades of a rotating fan or with another rotating (heavy) object which is not significantly influenced by the (light) quantum particle. The Schrödinger operator H(!t) = H0 + V!t is explicitly time-dependent. In this paper we continue the investigation of [1] and address mainly two questions: (i) existence of a unitary propagatorU(t; t0) which describes the time evolution of the system, (ii) the limit of rapid rotation where we show that the time evolution is well approximated by the evolution with the rotational invariant average potential. Applications to scattering theory will be treated in a subsequent paper. We will first introduce the model in more detail before we state the main results in Theorems 5.2 and 6.2. The coordinates are chosen in such a way that the rotation with constant angular velocity ! takes place in the x1; x2-plane, i.e., R(!t) = 0 os(!t) sin(!t) 0 sin(!t) os(!t) 0 0 0 1 21A : We denote by (x) the square integrable configuration space wave function of the (abstract) state in Hilbert space 2 H = L2(R ) and by ̂(p) its isometric Fourier transform, i.e., the momentum space wave function. The standard representation of this group 2000 Mathematics Subject Classification. 47A55, 47B25, 81Q15.