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


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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[1] ’Heavy ion Dynamics at intermediate energy’, Cyclotron Laboratory and Physics Department, Michigan State University, East Lansing, Michigan 48824, U.S.A., 1980.
[2] Linear Integral Operators (translated by ), Pitman Advanced Publishing Program, Boston-London-Melbourne, 1982.
[3] Perturbation Theory for Linear Operators, Springer Verlag, Berlin-Heidelberg-New York, 1966.
[4] , and , ’Quantum tunneling properties from a Wigner Function study’, Center for Solid State Electronics Research, Arizona State University, Tempe, AZ 85287, U.S.A., 1987.
[5] and , Lehrbuch der Theoretischen Physik III: Quantenmechanik, 2nd edn, Akademie Verlag, Berlin, 1966.
[6] and , ’An analysis of the quantum Liouville equation’, submitted, 1987.
[7] ’The nuclear Vlasov equation–methods and results that can (not) be taken over from the ”classical” case’, Proc. Workshop on Fluid Dynamical Approaches to the Many-Body Problem: Fundamental and Mathematical Aspects, Societa Italiana di Fisica, 1984.
[8] Private communication, 1987.
[9] Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York-Berlin-Heidelberg-Tokyo, 1983. · Zbl 0516.47023
[10] Communtation Properties of Hilbert Space Operators and Related Topics, Springer Verlag, Berlin-Heidelberg-New York, 1967.
[11] and , Methods of Modern Mathematical Physics II: Fourier Analysis, Academic Press, New York-London, 1972.
[12] and , Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, New York-San Francisco-London, 1975. · Zbl 0308.47002
[13] Simader, Math. Z. 138 pp 53– (1974)
[14] Simon, Arch. Rat. Mech. Anal. 52 pp 44– (1973)
[15] Tatarskii, Sov. phys. Usp. 26 pp 311– (1983)
[16] Wigner, Phys. Rev. 40 pp 749– (1932)
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