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The Schrödinger equation with a random potential. A mathematical review. (English) Zbl 0655.60050
Critical phenomena, random systems, gauge theories, Proc. Summer Sch. Theor. Phys., Sess. 43, Les Houches/France 1984, Pt. 2, 895-942 (1986).
[For the entire collection see Zbl 0651.00019.]
The paper represents a mathematical review of some results and problems in the quantum mechanics or spectral theory of random systems. A model which describes noninteracting electrons in a random potential (the so- called Anderson’s tight-binding model) is analyzed. The main object of consideration is the lattice Schrödinger operator \(H=-\Delta +v\) defined on the space of square summable complex valued functions on Z d, where \(\Delta\) denotes the finite difference Laplacian and v is a random potential.
The author investigates one of the most challenging questions concerning random Schrödinger operators, namely the time evolution of a wave packet \(\Psi_ t=e^{itH}\Psi_ o\). In particular, the phenomenon of localization (the wave packet does not spread) is discussed. Some of the theory considered by the author also applies to linear wave propagation in a random medium such as the transmission of light in a medium with a random diffractive index.
Reviewer: R.Manthey

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
81P20 Stochastic mechanics (including stochastic electrodynamics)
81Q15 Perturbation theories for operators and differential equations in quantum theory
35J10 Schrödinger operator, Schrödinger equation