×

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)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abramovitz, M., Stegun, I. A.: Handbook of mathematical functions. Dover: N.B.S. 1965
[2] Avron, Y., Herbst, I.: Spectral and scattering theory of Schr?dinger operators related to the Stark effect. Commun. Math. Phys.52, 239-254 (1977) · Zbl 0351.47007
[3] Carmona, R.: Exponential localization in one dimensional disorders systems. Duke Math. J.49, 191-213 (1982) · Zbl 0491.60058
[4] Craig, W., Simon, B.: Subharmonicity of the Liapunov index (to be published)
[5] Dunford, N., Schwarz, J. T.: Linear operators. II. New York: Wiley 1963 · Zbl 0128.34803
[6] Goldsheid, I. Ja., Mol?anov, S. A., Pastur, L. A.: A pure point spectrum of the stochastic one dimensional Schr?dinger equation. Funct. Anal. Appl.11, 1-10 (1977) · Zbl 0368.34015
[7] Herbst, I., Howland, J.: The Stark ladder and other one-dimensional external field problems. Commun. Math. Phys.80, 23 (1981) · Zbl 0473.47037
[8] H?rmander, L.: Hypoelliptic differential equations of second order. Acta Mathematica119, 147-171 (1967) · Zbl 0156.10701
[9] Kunz, H., Souillard, B.: Sur le spectre des op?rateurs aux diff?rences finies al?atoires. Commun. Math. Phys.78, 201-246 (1980) · Zbl 0449.60048
[10] Mol?anov, S. A.: The structure of eigenfunctions of one dimensional unordered structures. Math. USSR Izv.12, 69-101 (1978) · Zbl 0401.34023
[11] Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391-408 (1981) · Zbl 0489.47010
[12] Stone, M. H.: Linear transformations in Hilbert space and their applications to analysis. Providence: Am. Math. Soc. Coll. Publ.15, 1932 · Zbl 0005.40003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.