Abstract
It is shown that at high frequencies matrix elements of the Green's function of a random discrete wave equation decay exponentially at long distances. This is the input to the proof of dense point spectrum with localized eigenfunctions in this frequency range. The proof uses techniques of Fröhlich and Spencer. A sequence of renormalization transformations shows that large regions where wave propagation is easily maintained become increasingly sparse as resonance is approached.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Anderson, Absence of diffusion in certain random lattices,Phys. Rev. 109:1492–1505 (1958).
P. Wölfle and D. Vollhardt, Self-consistent diagrammatic treatment of Anderson localization, in Y. Nagaoka and H. Fukuyama, eds.,Anderson Localization (Springer, New York, 1982), pp. 26–43.
R. Carmona, Random Schrödinger operators, inÉcole d'Eté de Probabilités de Saint Flour XIV—1984, Lecture Notes in Mathematics 1180 (Springer, New York, 1986).
J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Commun. Math. Phys. 88:151–184 (1983).
F. Martinelli and H. Holden, On absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on L2(R v),Commun. Math. Phys. 93:197–217 (1984).
M. Stephen, Waves in disordered media, in The Mathematics and Physics of Disordered Media, Lecture Notes in Mathematics 1035 (Springer, Berlin, 1983), pp. 400–403.
W. Faris, A localization principle for multiplicative perturbations,J. Funct. Anal. 67:105–114 (1986).
W. Faris, Localization for a random discrete wave equation, inRandom Media, IMA Volumes in Mathematics and its Applications, Vol. 7 (Springer), to appear.
W. Faris,Self-Adjoint Operators, Lecture Notes in Mathematics 433 (Springer, New York, 1975).
B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians,Commun. Pure Appl. Math. 39:75–90 (1986).
F. Wegner, Bounds on the density of states in disordered systems,Z. Phys. B 44:9–15 (1981).
B. Simon, private communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Faris, W.G. Localization estimates for a random discrete wave equation at high frequency. J Stat Phys 46, 477–491 (1987). https://doi.org/10.1007/BF01013369
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01013369