Abstract
The purpose of this paper is to study a limit probability distribution of the set of the first κ eigenvalues λ1(ℒ)<λ2(ℒ)<...<λκ(ℒ) (with a fixed κ and ℒ→∞) of the boundary problem on the interval [0, ℒ]
wherea(t, ω),q(t, ω) are the random stationary processes. Particularly the question of the repulsion between the first eigenvalues (small energetic levels) is studied. It has been proved that in the “divergent” case (q(t, ω)=0,a(t, ω)≠0) levels repulsion exists. As for the “potential” case (a(t, ω)≡1,q(t, ω)≠0) there is not any repulsion at all. This is one of the main differences between these two cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Pastur, L.A.: The spectra of the random self-adjointness operators. Usp. Mat. Nauk28, 3–64 (1969), (1973). English transl. in Russian Math. Surv.28, (1973)
Kozlov, S.M.: The averaging of the random operators. Mat. Sb. 109 (151) 188–202 (1979)
Lipšic, I.M., Gredeskul, S.A., Pastur, L.A.: An introduction to disordered system theory. Moskva: Nauka 1982
Molčanov, S.A.: On the nature of the eigenfunctions of one-dimensional disordered structures. Izv. Akad. Nauk SSSR42, 70–103 (1978); English transl. in Math. SSSR Izv.12, (1978)
Molčanov, S.A.: The local structure of the spectrum of the Random one-dimensional Schrödinger operator. Commun. Math. Phys.78, 429–446 (1981)
Malkin, V.M.: The distribution of the distances between the energetic levels of the particle in one-dimensional random field. Preprint 81-49, IIAF SO Akad. Nauk SSSR, Novosibirsk, 1981
Casati, G., Valz-Gris, F., Guarniery, I.: On the connection between quantization of nonintegrable systems and statistical theory of spectra. Let. Nuovo Cimento28, 279 (1980)
Gordin, M.I., Lifšic, B.A.: The central limit theorem for Markov stationary processes. Dokl. Akad. Nauk SSSR239, 766–767 (1978)
Ibragimov, J.A., Linnik, Ju.V.: Independent and stationary connected variables. Moskva: Nauka 1965
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Narimanjan, S.M.: The central limit theorem for random walk on the circle. Dokl. Akad. Nauk Arm. SSSR64, 129–136 (1977)
Anshelevich, V.V., Vologodskii, A.V.: Laplace operator and Random walk on one-dimensional nonhomogeneous lattice. J. Stat. Phys.25, 419–430 (1981)
Anshelevich, V.V., Khanin, K.M., Sinai, Ya.G.: Symmetric Random walks in Random environments. Commun. Math. Phys.85, 449–470 (1982)
Friedrichs, K.O.: Perturbation of spectra in Hilbert space. Providence, Rhode Island: Am. Math. Society, 1965
Feller, W.: An introduction to probability theory and its applications. New York: Wiley 1966
Author information
Authors and Affiliations
Additional information
Communicated by Ya. G. Sinai
Rights and permissions
About this article
Cite this article
Grenkova, L.N., Molčanov, S.A. & Sudarev, J.N. On the basic states of one-dimensional disordered structures. Commun.Math. Phys. 90, 101–123 (1983). https://doi.org/10.1007/BF01209389
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01209389