Abstract
We consider the integrated density of states,k(E), of a general operator on ℓ2(ℤv) of the formh=h 0+v, where\((h_0 u)(n) = \sum\limits_{\left| i \right| = 1} {u(n + i)} \) and (vu)(n)=v(n)u(n), wherev is a general bounded ergodic stationary process on ℤv. We show that |k(E)−k(E′)|≦C[−log(|E−E′|]−1 when |E−E′|≦1/2, The key is a “Thouless formula for the strip.”
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Avron, J., Craig, W., Simon, B.: Large coupling behavior of the Lyaponov exponent for tight binding one dimensional random systems. J. Phys (submitted)
Avron, J., Deift, P., Spencer, T.: (unpublished)
Avron, J., Simon, B.: Almost periodic Schrödinger operators: I. Limit periodic potentials. Commun. Math. Phys.82, 101–120 (1982)
Avron, J.: Simon, B.: Almost periodic Schrödinger operators: II. The density of states. Duke Math. J. (to appear)
Benderskii, M., Pastur, L.: On the spectrum of the one dimensional Schrödinger equation with a random potential. Mat. Sb.82, 245–256 (1970)
Craig, W.: Pure point spectrum for discrete almost periodic Schrödinger operators. Commun. Math. Phys.88, 113–131 (1983)
Craig, W., Simon, B.: Subharmonicity of the Lyaponov index. Duke Math. J (submitted)
Guivarch, Y.: (private communication)
Kotani, S.: Lyaponov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. Proc. Kyoto Stoch. Conf. 1982
Pastur, L.: Spectral properties of disordered systems in one-body approximation. Commun. Math. Phys.75, 179 (1980)
Pöschel, J.: Examples of discrete Schrödinger operators with pure point spectrum. Commun. Math. Phys.88, 447–463 (1983)
Schmidt, H.: Disordered one-dimensional crystals. Phys. Rev.105, 425 (1957)
Simon, B.: Kotani theory for one dimensional stochastic Jacobi matrices. Commun. Math. Phys.89, 227–234 (1983)
Simon, B.: Trace ideals and their applications, Cambridge: Cambridge University Press 1979
Spencer, T.: (private communication)
Thomas, L.: Time dependent approach to scattering from impurities in a crystal. Commun. Math. Phys.33, 335–343 (1973)
Thouless, D.: A relation between the density of states and range of localization for one dimensional random systems. J. Phys. C5, 77–81 (1972)
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Communicated by T. Spencer
Also at Department of Physics, Research partially supported by USNSF Grant MCS-81-20833
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Craig, W., Simon, B. Log hölder continuity of the integrated density of states for stochastic Jacobi matrices. Commun.Math. Phys. 90, 207–218 (1983). https://doi.org/10.1007/BF01205503
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DOI: https://doi.org/10.1007/BF01205503