Non-differentiable Bohmian trajectories

Abstract

A solution ψ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field ( ψ/mψ) . In the case of one specific non-differentiable weak solution Ψ we show how Bohmian trajectories can be obtained for Ψ from the trajectories of a sequence Ψn → Ψ. (For any real t the sequence Ψn (t, •) converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors Ψ in the Hilbert space on which the Schrödinger evolution Ψ → e −iht Ψ acts.