We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schrödinger operators with nonmonotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.where the entries v(n) of the potential are random and independent.The basic phenomenon, named Anderson localisation after the physicist P. W. Anderson, is that disorder can cause localisation of electron states, which manifests itself in time evolution (non-spreading of wave packets), (vanishing of) conductivity in response to electric field, Hall currents in the presence of both magnetic and electric field, and statistics of the spacing between nearby energy levels. The first property implies spectral localisation, i.e. the spectral measure of H A is almost surely pure point, and almost sure exponential decay of eigenfunctions.These properties are known to hold for H A in each of the following cases: 1) high disorder (the coupling constant g is large), 2) extreme energies, 3) weak disorder away from the spectrum of the unperturbed operator, and 4) one dimension, d = 1.Historically, the first proof of spectral localisation was given by Goldsheid, Molchanov and Pastur [12], for a one-dimensional continuous random Schrödinger operator.In higher dimension, the absence of diffusion was first established in 1983 by Fröhlich and Spencer [9] using multi-scale analysis. Their approach has led to a multitude of results on localisation for a wide range of problems. The reader is referred to the monograph of Stollmann [17] or the recent lecture notes of W. Kirsch [14] for a review of the history of the subject and a gentle introduction to the multiscale analysis -which is not used here.One of the ingredients of multi-scale analysis is the regularity of the integrated density of states, the (distribution function of the) average of the spectral measure over the randomness.Ten years later Aizenman and Molchanov [2] introduced an alternative method for the proof of localisation, known as the fractional moment method, which has also found numerous applications. In particular, in [1], Aizenman introduced the notion of eigenfunction correlator, which, combined with the fractional moment method, allowed him to give the first proof of dynamical localisation. We refer to the lecture notes of Stolz [19] and Aizenman and Warzel [3] for a survey of subsequent developments.In the fractional moment method, an a priori estimate on the diagonal elements of the resolvent (H A − λ) −1 plays a key rôle in the underlying analysis.

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.

Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schr\"odinger operator on the complete graph. The operators exhibits local quasi modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the $\ell^1$-though not in $\ell^2$-sense, where the eigenvalues have the statistics of \v{S}eba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz-Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes

We study the almost Mathieu operator at critical coupling. We prove that there exists a dense G δ set of frequencies for which the spectrum is of zero Hausdorff dimension.