The three‐dimensional wigner‐poisson problem: Existence, uniqueness and approximation

We prove the existence of a unique, global classical solution of the quantum Vlasov–Poisson problem posed on the phase space ℝ   x 3× ℝ   v 3 . The proof is based on a reformulation of the quantum Vlasov–Poisson problem as a system of countably many Schrödinger equations coupled to a Poisson equation for the potential. The Schrödinger‐Poisson problem is first analysed on a bounded domain in ℝ   x 3and the solution of the whole‐space problem is then obtained by a limiting procedure in which the domains ‘tend’ to ℝ   x 3 .