Schrödinger operators with an electric field and random or deterministic potentials. (English) Zbl 0531.60061

We prove that the Schrödinger operator \(H=-d^ 2/dx^ 2+V(x)+F\cdot x\), on \(L^ 2(R)\), (\(F\neq 0)\) has purely absolutely continuous spectrum if V(x) is a bounded real-valued function whose first derivative is bounded, and V” is essentially bounded. If \(F=0\), and V is a random potential made of random wells of independent depth, the spectrum is almost surely pure point. Further results by Ben-Aztzi (Technion preprint) show that absolute continuity for \(F\neq 0\) is obtained when V has some integrable singularities. Nevertheless some regularity is necessary, as Delyon, B. Simon and B. Souillard prove (Caltech preprint) that in the case V is the sum of regularly spaced Dirac-delta functions with random coefficients and F is sufficiently small, the spectrum is almost surely pure point.


60H25 Random operators and equations (aspects of stochastic analysis)
81P20 Stochastic mechanics (including stochastic electrodynamics)
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