Abstract
We prove that the Schrödinger operatorH=−d 2/dx 2+V(x)+F·x has purely absolutely continuous spectrum for arbitrary constant external fieldF, for a large class of potentials; this result applies to many periodic, almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenF=0, the spectrum is almost surely pure point with exponentially decaying eigenfunctions.
Similar content being viewed by others
References
Abramovitz, M., Stegun, I. A.: Handbook of mathematical functions. Dover: N.B.S. 1965
Avron, Y., Herbst, I.: Spectral and scattering theory of Schrödinger operators related to the Stark effect. Commun. Math. Phys.52, 239–254 (1977)
Carmona, R.: Exponential localization in one dimensional disorders systems. Duke Math. J.49, 191–213 (1982)
Craig, W., Simon, B.: Subharmonicity of the Liapunov index (to be published)
Dunford, N., Schwarz, J. T.: Linear operators. II. New York: Wiley 1963
Goldsheid, I. Ja., Molčanov, S. A., Pastur, L. A.: A pure point spectrum of the stochastic one dimensional Schrödinger equation. Funct. Anal. Appl.11, 1–10 (1977)
Herbst, I., Howland, J.: The Stark ladder and other one-dimensional external field problems. Commun. Math. Phys.80, 23 (1981)
Hörmander, L.: Hypoelliptic differential equations of second order. Acta Mathematica119, 147–171 (1967)
Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys.78, 201–246 (1980)
Molčanov, S. A.: The structure of eigenfunctions of one dimensional unordered structures. Math. USSR Izv.12, 69–101 (1978)
Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981)
Stone, M. H.: Linear transformations in Hilbert space and their applications to analysis. Providence: Am. Math. Soc. Coll. Publ.15, 1932
Author information
Authors and Affiliations
Additional information
Communicated by T. Spencer
Partially supported by N.S.F. Grant MCS-82-02045
Partially supported by N.S.F. Grant MCS-81-20833
Rights and permissions
About this article
Cite this article
Bentosela, F., Carmona, R., Duclos, P. et al. Schrödinger operators with an electric field and random or deterministic potentials. Commun.Math. Phys. 88, 387–397 (1983). https://doi.org/10.1007/BF01213215
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01213215