977 0022-4715/04/0500-0977/0 © 2004 Plenum Publishing Corporation Journal of Statistical Physics, Vol. 115, Nos. 3/4, May 2004 (© 2004) Asymptotic Behavior of the Integrated Density of States of Acoustic Operators with Random Long Range Perturbations Hatem Najar 1 1 Laboratoire Analyse, Géometrie et Application, Institut Galillée, Université de Paris 13, 93430 Villetaneuse, France; e-mail: najar@math.univ-paris13.fr Received August 13, 2003; accepted December 3, 2003 In this paper we study the behavior of the integrated density of states of random acoustic operators of the form A w =-N 1 +w N. When + w is considered as an Anderson type long range perturbations of some periodic function, the behavior of the integrated density of states of A w in the vicinity of the internal spectral edges is given. KEY WORDS: Spectral theory; random operators; integrated density of states; Lifshitz tails. 1. INTRODUCTION Basic properties of wave propagation in a nonhomogeneous medium even- tually boil down to the spectral properties of the relevant self-adjoint dif- ferential operator. As far as the acoustic waves are concerned, they are governed by the so called ‘‘acoustic operator.’’ (2) It is a self adjoint opera- tor on L 2 (R d ) and formally defined by: A w =A(+ w )=-N 1 + w N=- C d i=1 “ x i 1 + w “ x i , (1.1) where + w is a positive and bounded function which represents the mass density or the elasticity contrast of the medium where the wave propagates. See ref. 2 and references therein for more physicals interpretations and motivation.