An elementary proof is given of localization for linear operators H=H o +lV, with H o translation invariant, or periodic, and V( . ) a random potential, in energy regimes which for weak disorder (l®0) are close to the unperturbed spectrum s(H o ). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [AM]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements <0|P [a,b] e -itH |x> of the spectrally filtered unitary time evolution operators, with [a,b] in the relevant energy range. corrected 7/12/93 Localization at Weak Disorder 2 1. Introduction This work presents an elementary derivation of localization for time evolutions generated by linear operators consisting of a translation invariant, or periodic, part and an added random potential, at energy rang...

A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient `Lifshitz tail estimates' on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the "multisca...

The spreading of wave packets evolving under the Anderson Hamiltonian on the Bethe Lattice is studied for small disorder. The mean square distance travelled by a particle in a time t is shown to grow as t 2 for large t. 1 INTRODUCTION The Anderson model [6] gives a description of the motion of a quantum-mechanical electron in a crystal with impurities. It is given by the random Schrodinger operator H = 1 2 \Delta + V on ` 2 (L ) ; (1.1) where L is either Z d or the Bethe lattice B (same as Cayley tree - an infinite connected graph with no closed loops and a fixed number K + 1 of nearest neighbors at each vertex (K 2, so B is not the line R ); the distance between two sites x and y in B will be denoted by d(x; y) and is equal to the length of the shortest path connecting x and y.) The (centered) Laplacian \Delta is defined by (\Deltau)(x) = X y u(y) ; (1.2) To appear in Communications in Mathematical Physics. y 1991 Mathematics Subject Classification. Primary 82B44. ...

We study the localization of wave functions for one-dimensional Schrödinger Hamiltonians with random potentials V (x) with short range correlations and large local fluctuations such that ∫ dx 〈V (x)V (0) 〉 = ∞. A random supersymmetric Hamiltonian is also considered. Depending on how large the fluctuations of V (x) are, we find either new energy dependences of the localization length, ℓloc ∝ E / lnE, ℓloc ∝ E µ/2 with 0 < µ < 2 or ℓloc ∝ ln µ−1 E for µ> 1, or superlocalization (decay of the wave functions faster than a simple exponential). PACS numbers: 72.15.Rn; 73.20.Fz; 02.50.-r. Introduction. – The phenomenon of Anderson localization [1] in one dimension has been widely studied since the pioneering work of Mott & Twose arguing that all states are localized in onedimension (1d) [2]. This statement was rigorously proven in Refs. [3, 4]. A general method to study the spectral and localization properties of 1d random Hamiltonian was proposed in Refs. [5, 6] : let us consider the one-dimensional Schrödinger Hamiltonian H = − d2

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