The limit from the Schrödinger‐Poisson to the Vlasov‐Poisson equations with general data in one dimension

We deal with the classical limit of the Schrodinger-Poisson system to the Vlasov-Poisson equations as the Planck constant E goes to zero. This limit is also frequently called the semiclassical limit. The coupled Schrodinger-Poisson system for the wave functions {ψ∈ j (t, x)} is transformed to the Wigner-Poisson equations for a phase space function f∈(t, x, ξ). For the case of the so-called completely mixed state, i.e., j = 1, 2,…,∞, under additional assumptions on the potential, this classical limit was solved in 1993 by P.-L. Lions and T. Paul in [23] and, independently, by P. A. Markowich and N. J. Mauser in [26] with strong assumptions on the initial data. The so-called pure state case, where only one or a finite number of wave functions {ψ∈ j (t, x)} are considered, has been open up to now. We prove here for general initial data (the pure-state as well as the mixed-state case) of the wave functions in one space dimension that the Wigner measure f (t, x, ξ), which is a weak limit of f∈ (t, x, ξ) as ∈ tends to 0, satisfies the classical one-dimensional Vlasov-Poisson equations. As a by-product, we have improved the decay assumption on the initial data of one-dimensional Vlasov-Poisson equations in [38] for the existence of global weak solutions with measures as initial data. The equations we regard are widely used in quantum/classical transport and semiconductor theory as a nonlinear one-particle (mean field) approximation of the linear N-electron Schrodinger/Hamilton equation.