Abstract
We investigate the spectrum of Schrödinger operatorsH ω of the type:H ω=−Δ+∑q i (ω)f(x−x i +ζ i (ω))(q i (ω) and ζ i (ω) independent identically distributed random variables,i∈ℤd). We establish a strong connection between the spectrum ofH ω and the spectra of deterministic periodic Schrödinger operators. From this we derive a condition for the existence of “forbidden zones” in the spectrum ofH ω. For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Albeverio, S., Fenstad J. E., Høegh-Krohn, R.: Singular perturbations and nonstandard analysis. Tran. Am. Math. Soc.252, 275–295 (1979)
Albeverio, S., Høegh-Krohn, R.: Point interactions as Limits of short range interaction, Preprint, Bochum, 1980
Avron, J. E., Grossmann, A., Rodriguez, R.: Spectral properties of reduced Bloch Hamiltonians. Ann. Phys.103, 47–63 (1977)
Borland, R. E.: Existence of energy gaps in one-dimensional liquids. Proc. Phys. Soc. (London)78, 926 (1960)
Faris, W. G.: Self-adjoint operators. In: Lecture Notes in Mathematics Vol.433, Berlin: Springer 1975
Flügge, S.: Rechenmethoden der Quantenmechanik, Berlin: Springer 1965
Frisch, H. L., Lloyd, S. P.: Electron levels in a one-dimensional random lattice. Phys. Rev.120, 1175–1189 (1960); reprinted in [14]
Fakushima, M., Nagai, H., Nakao, S.: On an asymptotic property of spectra of random difference operator, Proc. Jpn. Acad.51, 100–102 (1975)
Grossman, A., Høegh-Krohn, R., Mebkhout, M.: The one particle theory of point interactions; Commun. Math. Phys.77, 87–110 (1980)
Halperin, B. I.: Properties of a particle in a one-dimensional random potential. Adv. Chem. Phys.13, 123–178 (1967)
Kirsch, W., Martinelli, F.: On the ergodic properties of the spectrum of general random operators. Preprint, Bochum, 1980
Kronig, R. de L., Penney, W. G.: Quantum mechanics of electrons in crystal lattices. Proc. R. Soc. (London)A130, 499–513 (1931); reprinted in [14]
Kunz, H.; Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires, Commun. Math. Phys.78, 201–246 (1980)
Lieb, E. H., Mattis, D. C.: Mathematical physics in one dimension. New York: Academic Press 1966
Luttinger, J. M.: Wave propagation in one-dimensional structure. Philips Res. Rep.6, 303–310 (1951); reprinted in [14]
Morgan, J. D. III: Schrödinger operators whose potentials have separated singularities. J. Operat. Theory1, 109–115 (1979)
Nakao, S.: On the spectral distribution of the Schrödinger operator with random potential. Jpn. J. Math.3, 111–139 (1977)
Pastur, L. A.: Spectra of random selfadjoint operators. Russ. Math. Surv.28, 1–67 (1973)
Pastur, L. A.: Spectral properties of disordered systems in the one-body approximation. Commun. Math. Phys.75, 179–196 (1980)
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. I, II, IV. New York: Academic Press 1978
Saxon, D. S., Hutner, R. A.: Some electronic properties of a one-dimensional crystal model. Philips Res. Rep.4, 81–122 (1949)
Weidmann, J.: Lineare Operatoren in Hilberträumen. Stuttgart: Teubner 1976 (english translation: Linear operators in Hilbert spaces Berlin: Springer 1980)
Author information
Authors and Affiliations
Additional information
Communicated by Ya. G. Sinai
Rights and permissions
About this article
Cite this article
Kirsch, W., Martinelli, F. On the spectrum of Schrödinger operators with a random potential. Commun.Math. Phys. 85, 329–350 (1982). https://doi.org/10.1007/BF01208718
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01208718