## 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.

### MSC:

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

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