On the weak coupling limit of quantum manybody dynamics and the quantum Boltzmann equation
Xuwen Chen  Department of Mathematics, University of Rochester, Rochester, NY 14627, United States (email) Abstract: The rigorous derivation of the UehlingUhlenbeck equation from more fundamental quantum manyparticle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum $N$ particle dynamics. We assume the integral of the microscopic interaction is zero and we assume $W^{4,1}$ perparticle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weakcoupling limit must satisfy the infinite quantum MaxwellBoltzmann hierarchy instead of the expected infinite UehlingUhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the UehlingUhlenbeck equation, one must work with perparticle regularity bound below $W^{4,1}$.
Keywords: Quantum manybody dynamics, quantum Boltzmann equation, BBGKY hierarchy, UehlingUhlenbeck equation, weak coupling.
Received: December 2014; Revised: February 2015; Available Online: June 2015. 
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