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

Abstract

We deal with the classical limit of the Schrödinger-Poisson system to the Vlasov-Poisson equations as the Planck constant goes to zero. This limit is also frequently called the "semiclassical limit." The coupled Schrödinger-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. 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 mixedstate 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 Schrödinger/Hamilton equation.

Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms

“…In [33] the classical limit of the one dimensional S-P to the V-P system for general initial data; in particular the "pure state case", without assumption (2.16), was given. The trick in the pure state case is to change the concept of solutions of V-P such that jumps in ∇V are allowed and A convergence of the Wigner measure f is enough.…”

“…For the "pure state" result below this non-uniqueness of the limiting Wigner measure corresponds to the non-uniqueness of the kind of solutions of V-P introduced in [34] and used in [33]. See [22] for a construction of non-unique solutions to 1-d V-P (in a setting which is periodic in x, however) where different "regularization schemes" give different limits (cf also the remarks in [35]).…”

“…2) By the proof of Theorem 4.2 in [34] and Appendix B of [33], we can still prove the global existence of weak solution to (2.29) with f I (x, ξ) ∈ M + (R 2 ) and (2.24) holds only for some α > 0.…”

“…Before the presentation of 1-d pure state result of [33], let us first recall some related existence results for V-P, i.e. (1.14), (1.15) with b = 0 andṼ being the Coulomb potential.…”

The Vlasov–poisson Equations as the Semiclassical Limit of the Schrödinger–poisson Equations: A Numerical Study

“…The convergence of the limit was studied in [8,11,13,17]. For convenience, we summarize the main results of [17] as below.…”