ar X iv : m at h - ph / 9 90 80 09 v 1 6 A ug 1 99 9 Schrödinger Equation with Moving Point Interactions in Three Dimensions

We consider the motion of a non relativistic quantum particle in R 3 subject to n point interactions which are moving on given smooth trajec-tories. Due to the singular character of the time-dependent interaction, the corresponding Schrödinger equation does not have solutions in a strong sense and, moreover, standard perturbation techniques cannot be used. Here we prove that, for smooth initial data, there is a unique weak solution by reducing the problem to the solution of a Volterra integral equation involving only the time variable. It is also shown that the evolution operator uniquely extends to a unitary operator in L 2 (R 3).