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On the equivalence of the Schrödinger and the quantum Liouville equations. (English) Zbl 0696.47042
The author treats the quantum Liouville equation \[ \partial w/\partial t+m^{-1}\vec v\cdot \text{grad}_{\vec x}w=\Theta (\vec x,\partial_{\vec v})w=0,\quad t\in {\mathbb{R}}, \] in the phase-space \({\mathbb{R}}^ d\times {\mathbb{R}}^ d\), where \(\vec x\in {\mathbb{R}}^ d\) denotes position, \(\vec v\in {\mathbb{R}}^ d\) denotes momentum, m is particle mass, and \(\Theta\) (\(\vec x,\partial_{\vec v})\) is the pseudo-differential operator \((1/ih)[\psi (\vec x+(ih/2)\partial_{\vec v})-\psi (\vec x- (ih/2)\partial_{\vec v})],\) \(\psi\) \(=\) the potential energy. The equation governs the evolution of the quasi-distribution w(\(\vec x,\vec v,t)\) of an electron ensemble under the action of a scalar potential \(\psi\). The author proves that this quantum Liouville operator generates a strongly continuous group of operators if the Hamiltonian \(H=-(h^ 2/2m)\Delta +\psi\) is essentially selfadjoint in \(L^ 2({\mathbb{R}}^ d)\). The equation is applicable to the simulation of submicron semi-conductor devices, whose performance relies upon tunnelling effects.
Reviewer: G.F.Webb

MSC:
47D03 Groups and semigroups of linear operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82C70 Transport processes in time-dependent statistical mechanics
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