Localization and Intermittency: New Results

Here A is the Laplacian (continuous or on lattice) and V(x,w) is the random homogeneous field (or process) on a probability space (Q,F,P). The central achievement of one-dimensional theory was a series of S. Kotani's articles [l]-[3], where he discovered deep connections between Ljapunov's exponents y(X), the structure of the prediction of the potential V(-) and the spectral localization of H. Techniques of the cluster resolvent expansions, developed by Fröhlich and Spencer [4], together with generalization of Kotani's idea [2], proposed almost simultaneously (in different forms by Souillard et al. [5] and Simon-Wolff [6]) made it possibile to solve the point spectrum problem for Anderson's tightbinding model (1) in the multidimensional lattice case Z, d > 1. These results are summed up in the review article of Martinelli-Scoppola [7]. Simplifications and generalizations were proposed recently in this field by Dreifus and Klein [8]. Probabilistic approach, nevertheless, did not clear up the central physical idea of localization—the absence of resonance between a quantum particle with a given admissible energy X and some (rich enough) family of the blocks of the potential. It's natural to attempt to formulate the direct geometric conditions for the individual potential V(-), which will lead to the localization of the spectrum. The first and a very important step in this direction was made in the paper Simon-Spencer [9]. They proved (in the one-dimensional case) that unboundness of potential V gives us the singular spectrum. Moreover, in a multidimensional situation (d > 1), even in the lattice case, there are many open problems on the thin structure of the spectrum H, and on the existence of bifurcations with respect to some parameters of the model, etc. It's related, for example, to the simplest functional of o(H), namely the integral density of states N(X) (the so-called problem of Lifshitz tails). In this lecture I'll give an account of some recent results in this direction. It is based on the papers, which I wrote together with my friends and colleagues, especially J. Görtner, A. Gordon, L. Pastur and B. Simon. This lecture was prepared at the time of my visit to Caltech (spring 1990) and I am very grateful to Barry Simon for his hospitality, to A. Klein, R. Carmona,