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The three‐dimensional wigner‐poisson problem: Existence, uniqueness and approximation
Author(s) -
Brezzi Franco,
Markowich Peter A.
Publication year - 1991
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670140103
Subject(s) - mathematics , uniqueness theorem for poisson's equation , uniqueness , poisson distribution , bounded function , phase space , poisson's equation , space (punctuation) , domain (mathematical analysis) , mathematical analysis , quantum , limiting , schrödinger's cat , quantum mechanics , physics , linguistics , philosophy , mechanical engineering , engineering , statistics
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 .