In this paper we prove that the one dimensional Schrόdinger operator on 1 2 (Z) with potential given by:has a Cantor spectrum of zero Lebesgue measure for any irrational α and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all xeT.

We prove that, for large disorder or near the band tails, the spectrum of the Anderson tight binding Hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues. The corresponding eigenfunctions are exponentially well localized. These results hold in arbitrary dimension and with probability one. In one dimension, we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder. Our techniques extend to other physical systems which exhibit localization phenomena, such as infinite systems of coupled harmonic oscillators, or random Schrodinger operators in the continuum.

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In this paper we study metastability and nucleation for a local version of the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let ␤Ͼ0 be the inverse temperature and let ⌳ ʚ⌳ ␤ ʚZ 2 be two finite boxes. Particles perform independent random walks on ⌳ ␤ ‫⌳گ‬ and inside ⌳ feel exclusion as well as a binding energy UϾ0 with particles at neighboring sites, i.e., inside ⌳ the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that ⌳ is empty, while a total of ͉⌳ ␤ ͉ particles is distributed randomly over ⌳ ␤ ‫⌳گ‬ with no exclusion. That is to say, initially the system is in equilibrium with particle density conditioned on ⌳ being empty. For large ␤, the system in equilibrium has ⌳ fully occupied because of the binding energy. We consider the case where ϭe Ϫ⌬␤ for some ⌬ (U,2U) and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet and the time of its creation in the limit as ␤→ϱ for fixed ⌳ and lim ␤→ϱ (1/␤) log͉⌳ ␤ ͉ϭϱ. In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice ⌬ (U,2U) corresponds to the situation where the critical droplet has side length l c (1,ϱ), i.e., the system is metastable. The side length of ⌳ must be much larger than l c and independent of ␤, but is otherwise arbitrary. Because particles are conserved under Kawasaki dynamics, the analysis of metastability and nucleation is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas in ⌳ ␤ ‫⌳گ‬ can be treated as a reservoir that creates particles with rate at sites on the interior boundary of ⌳ and annihilates particles with rate 1 at sites on the exterior boundary of ⌳ . Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside ⌳ in the presence of a nonconservative boundary. The dynamics inside ⌳ is still conservative and this difficulty has to be handled via local geometric arguments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For instance, rectangular droplets tend to become square through a movement of particles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are attracted by the maximal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions ͑at least for not too long a time͒ without being attracted by a square.

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