Localization In The Anderson Model With Long Range Hopping

We give a proof of exponential localization in the Anderson model with long range hopping based on a multiscale analysis. 1 INTRODUCTION We consider the random Hamiltonian H = \Gamma + V on ` 2 (Z d ) ; (1.1) where 1. \Gamma is a translation invariant self-adjoint operator with exponentially decaying matrix elements, i.e., \Gamma(x; y) = OE(x \Gamma y) for some function OE on Z d with OE(\Gammax) = OE(x) for which there exist C ! 1 and fl ? 0 such that j\Gamma(x; y)j = jOE(x \Gamma y)j C e \Gammafl kx\Gammayk (1.2) for all x; y 2 Z d . Partially supported by the NSF under grant DMS-9208029. y To appear in the Brazilian Journal of Physics 23, 367-371 (1994) 2. V (x) ; x 2 Z d , are independent identically distributed random variables with common probability distribution ¯. In the usual Anderson model [1] \Gamma = \Gamma\Delta, where \Delta(x; y) = 1 if jx \Gamma yj = 1 and zero otherwise. In this article we are concerned with localization. We say that the random ..