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Localization at large disorder and at extreme energies: an elementary derivation. (English) Zbl 0782.60044
Summary: The work presents a short proof of localization under the conditions of either strong disorder \((\lambda >\lambda_ 0)\) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting in \(\ell^ 2\) spaces. A prototypical example is the discrete Schrödinger operator \(H=-\Delta+U_ 0(x)+\lambda V_ x\) on \(\mathbb Z^ d\), \(d\geq 1\), with \(U_ 0(x)\) a specified background potential and \(\{V_ x\}\) generated as random variables. The general results apply to operators with \(-\Delta\) replaced by a non-local self adjoint operator \(T\) whose matrix elements satisfy: \(\sum_ y| T_{x,y}|^ s\leq \text{Const}.,\) uniformly in \(x\), for some \(s<1\). Localization means here that within a specified energy range the spectrum of \(H\) is of the pure-point type, or equivalently – the wave functions do not spread indefinitely under the unitary time evolution generated by \(H\). The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elements \(T_{x,y}\). Under rapid decay of \(T_{x,y}\), the corresponding eigenfunctions are also proved to decay exponentially. The method is based on resolvent techniques.
The central technical ideas include the use of low moments of the resolvent kernel, i.e., \(\langle | G_ E(x,y)|^ s\rangle\) with \(s\) small enough \((<1)\) to avoid the divergence caused by the distribution’s Cauchy tails, and an effective use of the simple form of the dependence of \(G_ E(x,y)\) on the individual matrix elements of \(H\) in elucidating the implications of the fundamental equation \((H-E)G_ E(x,x_ 0)=\delta_{x,x_ 0}\). This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random part \(U_ 0(x)\), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms \((T_{x,y}\approx 1/| x- y|^{d+\varepsilon})\), and iv) localization produced by disordered boundary conditions.

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47B80 Random linear operators
47A10 Spectrum, resolvent
47N55 Applications of operator theory in statistical physics (MSC2000)
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[1] [A] Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev.109, 1492–1505 (1958) · doi:10.1103/PhysRev.109.1492
[2] [AALR] Abraham, E., Anderson, P.W., Licciardello, D.C., Ramakrishnan, T.V.: Scaling theory of localization: Absence of quantum diffusion in two dimensions. Phys. Rev. Lett.42, 673–676 (1979) · doi:10.1103/PhysRevLett.42.673
[3] [CFKS] Cycon, H., Froese, R., Kirsh, W., Simon, B.: Topics in the Theory of Schrödinger Operators. Berlin, Heidelberg, New York: Springer-Verlag 1987
[4] [CL] Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Boston: Birkhäuser 1990 · Zbl 0717.60074
[5] [CKM] Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys.108, 41 (1987) · Zbl 0615.60098 · doi:10.1007/BF01210702
[6] [D] von Dreifus, H.: On the effects of randomness in ferromagnetic models and Schrödinger operators. NYU Ph.. Thesis (1987)
[7] [DK1] von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys.124, 285–299 (1989) · Zbl 0698.60051 · doi:10.1007/BF01219198
[8] [DK2] von Dreifus, H., Klein, A.: Localization for random Schrödinger operators with correlated potentials. Commun. Math. Phys.140, 133–147 (1991) · Zbl 0734.60070 · doi:10.1007/BF02099294
[9] [DLS1] Delyon, F., Levy, Y., Souillard, B.: Anderson localization for multidimensional systems at large disorder or low energy. Commun. Math. Phys.100, 463–470 (1985) · Zbl 0576.60053 · doi:10.1007/BF01217724
[10] [DLS2] Delyon, F., Levy, Y., Souillard, B.: Anderson localization for one and quasi one-dimensional systems. J. Stat. Phys.41, 375 (1985) · doi:10.1007/BF01009014
[11] [F] Faris, W.: Localization estimates for off-diagonal disorder. Localization estimates for off-diagonal disorder. In Mathematics of Random Media, Lecture Notes in Appl. Mat.27, W.E. Kohler, B.J. White (eds.), pp. 391–406, Providence, R.I.: AMS 1991 · Zbl 0734.35163
[12] [FMSS] Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: A constructive proof of localization in Anderson tight binding model. Commun. Math. Phys.101, 21–46 (1985) · Zbl 0573.60096 · doi:10.1007/BF01212355
[13] [FP] Figotin, A., Pastur, L.: Spectral properties of disordered systems in the one-body approximation. Berlin, Heidelberg, New York: Springer-Verlag 1991
[14] [FS] Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys.88, 151–184 (1983) · Zbl 0519.60066 · doi:10.1007/BF01209475
[15] [G] Gross, L.: Decay of Correlations at High Temperature. Commun. Math. Phys.68, 9–27 (1979) · Zbl 0442.60097 · doi:10.1007/BF01562538
[16] [GJMS] Gordon, A., Jaksiĉ, V., Molchanov, S., Simon, B.: Spectral properties of random Schrödinger operators with unbounded potential. Commun. Math. Phys.
[17] [GMP] Goldsheid, I., Molchanov, S., Pastur, L.: A pure point spectrum of the one-dimensional Schrödinger operator. Funct. Anal. Appl.11, 1–10 (1977) · Zbl 0368.34015 · doi:10.1007/BF01135526
[18] [K] Kotani, S.: Lyapunov exponent and spectra for one-dimensional random Schrödinger operators. In: Proc. conf. on random matrices and their applic. Contemp. Math.50. Providence RI, 1986 · Zbl 0587.60054
[19] [KP] Kirsh, W., Pastur, L.: Bochum preprint (1990)
[20] [KS] Kunz, H., Souillard, B.: The localization transition on the Bethe lattice. J. Phys. (Paris) Lett.44, 411–414 (1983)
[21] [L] Landauer, R.: Phil. Mag.21, 863 (1970) · doi:10.1080/14786437008238472
[22] [MS] Molchanov, S., Simon B.: Localization theorem for non-local Schrödinger operator in one-dimensional lattice case. Caltech preprint
[23] [MT] Mott, N., Twose, W.: The theory of impurity conduction. Adv. Phys.10, 107–163 (1961) · doi:10.1080/00018736100101271
[24] [P] Pastur, L.A.: Random and almost periodic operators: New examples of spectral behaviour. In: IXth International Congress on Mathematical Physics, Simon, B., Truman, A., Davis, I.M. (eds.) Bristol, UK: Adam Hilger 1989
[25] [RS] Reed, M., Simon, B.: Methods of modern mathematical physics, vol.II (Fourier analysis, self-adjointness). New York: Acad. Press 1975 · Zbl 0308.47002
[26] [Si] Simon, B.: Localization in general one dimensional systems, I. Jacobi matrices. Commun. Math. Phys.102, 327–336 (1985) · Zbl 0604.60062 · doi:10.1007/BF01229383
[27] [Sp] Spencer, T.: Localization for random and quasiperiodic potentials. J. Stat. Phys.51, 1009–1019 (1988) · Zbl 1086.82547 · doi:10.1007/BF01014897
[28] [SiSp] Simon, B., Spencer, T.: Trance class perturbation and the absence of absolutely continuous spectrum. Commun. Math. Phys.125, 113–125 (1989) · Zbl 0684.47010 · doi:10.1007/BF01217772
[29] [SW] Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math.39, 75–90 (1986) · Zbl 0609.47001 · doi:10.1002/cpa.3160390105
[30] [T] Thouless, D.: Electrons in disordered systems and the theory of localization. Phys. Rev.13, 93–106 (1974)
[31] [W] Wegner, F.: Bounds of the density of states in disordered systems. Z. Phys. B. Condensed Matter44, 9–15 (1981) · doi:10.1007/BF01292646
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