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On absorbing boundary conditions for quantum transport equations. (English) Zbl 0821.45002

The Wigner equation is considered: \(w_ t + v \cdot \nabla_ x w + \Theta [V]w = 0,\) where \(w = w(x,v,t)\) is a real-valued function; \(x,v \in \mathbb{R}^ d\), \(d = 1,2\) or 3; \[ \Theta [V]w : = {i \over (2 \pi)^ d} \int_{\mathbb{R}^ d_ \eta} \int_{\mathbb{R}^ d_{v'}} \delta V(x, \eta, t)w(x,v',t) e^{i(v - v') \cdot \eta} dv'd \eta, \]
\[ \delta V(x, \eta,t) : = V \left( x + {\eta \over 2}, t \right) - V \left( x - {\eta \over 2}, t \right). \] The function \(w\) is called the Wigner function and is used for simulations of quantum devices. The author constructs the absorbing boundary conditions for (1) and analyzes the well-posedness of the corresponding boundary value problems.

MSC:

45K05 Integro-partial differential equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

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