Self-adjointness and the Existence of Deterministic Trajectories in Quantum Theory

We show that the particle motion in Bohmian mechanics as the solution of an ordinary differential equation exists globally, i.e., the singularities of the velocity field and infinity will not be reached in finite time for typical initial configurations and a large class of potentials, including the physically most relevant potential of N -particle Coulomb interaction with arbitrary charges and masses. The analysis is based on the probabilistic significance of the quantum flux. We point to the connection between the global existence of Bohmian mechanics and the self-adjointness of the Schrödinger Hamiltonian.