Boundary conditions and the Wigner equation solution
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We consider the existence and uniqueness of the solution of the Wigner equation in the presence of boundary conditions. The equation, describing electron transport in nanostructures, is analyzed in terms of the Neumann series expansion of the corresponding integral form, obtained with the help of classical particle trajectories. It is shown that the mathematical aspects of the solution can not be separated from the physical attributes of the problem. In the presented analysis these two sides of the problem mutually interplay, which is of importance for understanding of the peculiarities of Wigner-quantum transport. The problem is first formulated as the long time limit of a general evolution process posed by initial and boundary conditions. Then the Wigner equation is reformulated as a second kind of a Fredholm integral equation which is of Volterra type with respect to the time variable. The analysis of the convergence of the corresponding Neumann series, sometimes called Liouville–Neumann series, relies on the assumption for reasonable local conditions obeyed by the kernel.
KeywordsWigner equation Monte Carlo method Boundary conditions Convergence
This work is partially supported by the Bulgarian Science Fund under Grant DFNI I02/20, as well as by the EU AComIn Project FP7-REGPOT-2012-2013-1.
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