Abstract
We consider a class of long range Hamiltonians with diagonal disorder onl 2 (Z). For anyergodic potentialV with non-empty essential range, we prove the exponential decay of the Green's functions for energies in the essential range. IfV is independent identically distributed, we obtain the exponential decay of the Green's functions for all coupling constant λ>0. Moreover the Hamiltonian has only pure point spectrum.
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Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B.: Schrödinger Operators. Berlin, Heidelberg, New York: Springer 1987
Delyon, F., Kunz, H., Souillard, B.: One dimensional wave equations in disordered media. J. Phys. A16, 25 (1983)
Delyon, F., Lévy, Y., Souillard, B.: Anderson localization for one and quasi-one dimensional systems. J. Stat. Phys.41, 375–388 (1985)
Delyon, F., Lévy, Y., Souillard, B.: Anderson localization for multidimensional systems at large disorder or large energy. Commun. Math. Phys.100, 463 (1985)
Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat.31, 457–469 (1960)
Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys.84, 403–438 (1982)
Katznelson, Y.: An introduction to Harmonic analysis. New York: Dover 1976
Kotani, S.: Lyapunov indices determine absolutely continuous spectra of stationary random one dimensional Schrödinger operators. Proc. Kyoto Stoch. Conf. 225–248 (1982)
Kotani, S.: Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys.1, 129–133 (1989)
Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys.78, 201 (1980)
Ruelle, D.: Erogdic theory of differentiable dynamical systems. IHES50, 275 (1979)
Simon, B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys.89, 227 (1983)
Simon, B., Spencer, T.: Trace class perturbations and the absence of absolutely continuous spectra. Commun. Math. Phys.125, 113 (1989)
Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure. Appl. Math39, 75–90 (1986)
Spencer, T.: The Schrödinger equation with a random potential — a mathematical review. Critical Phenomena, Random Systems, Gauge Theories, Les Houches XLIII. Osterwalder, K., Stora, R. (eds.)
Thouless, D. J.: 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 B. Simon
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Wang, WM. Exponential decay of Green's functions for a class of long range Hamiltonians. Commun.Math. Phys. 136, 35–41 (1991). https://doi.org/10.1007/BF02096789
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DOI: https://doi.org/10.1007/BF02096789