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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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##### References:
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