Linear Schrödinger evolution equations with moving Coulomb singularities

The Cauchy problem for Schrödinger evolution equations with a finite number of moving Coulomb singularities is investigated. The case of a single singularity has been studied by several authors (see, e.g., Baudouin et al. (2005) [1], and Okazawa et al. (2010) [19], ). However, it seems to be no previous work on plural singularities. We shall show that the problem has a unique (classical) solution by using a time-dependent linear transformation of the unknown function which locally freezes the motion of the whole singularities under the simplest collisionless condition. In fact, a new existence and uniqueness theorem is available for the transformed problem. Such an abstract framework is established from the viewpoint of linear evolution equations of hyperbolic type in a Hilbert space as an innovative modification of those in Okazawa (1998) [17], and Okazawa and Yoshii (2011) [20].