F. Delyon

Absence of localization in a class of Schrödinger operators with quasiperiodic potential

We prove that a class of discrete Schrödinger operators with a quasiperiodic potential taking only a finite number of values, exhibits purely continuous spectrum; in particular they cannot have localized eigenvectors.

Anderson localization for multi-dimensional systems at large disorder or large energy

We prove that discrete Schrödinger operators on ℤd with a random-potential have almost-surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy. This is the first proof of localization for multi-dimensional Anderson models.

The rotation number for finite difference operators and its properties

We discuss a rotation number α(λ) for second order finite difference operators. Ifk(λ) denotes the integrated density of states, thenk(λ)=2α(λ). For almost periodic operators,k(λ) is proved to lie in the frequency-module whenever λ is outside the spectrum; this yields a labelling of the gaps of the spectrum.

Purely Absolutely Continuous Spectrum for Almost Mathieu Operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl2(ℤ), (lY,λ Ψ)(n) = Ψ(l+1)+(l−)+λ cos(ωn+α) Ψ(n) have a purely absolutely continuous spectrum for almost all a provided that ω is a good irrational and λ is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.

Anderson localization for one- and quasi-one-dimensional systems

We prove almost-sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete Schrödinger operators on one- and quasi-one-dimensional lattices. This paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems; we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential. Our results allow us to prove that, e.g., for strong disorder, the smallest positive Lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity.

Recurrence of the eigenstates of a Schrödinger operator with automatic potential

We consider the Schrödinger eigenvalue problem in the discrete case with a potential assuming two values distributed according to the automatic sequence of Prouhet-Thue-Morse. We show that there are no localized states and that the generalized eigenvectors are recurrent on a geometrical set stemming from the hierarchical nature of the potential.

Lower bounds on the cluster size distribution

We rigorously prove that the probabilityPn that the origin of ad-dimensional lattice belongs to a cluster of exactlyn sites satisfiesPn > exp(−αn(d−1)/d) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentrationp > pc, as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.

Appearance of a purely singular continuous spectrum in a class of random schrödinger operators

We consider a discrete Schrödinger operator on l2(ℤ) with a random potential decaying at infinity as ¦n¦−1/2. We prove that its spectrum is purely singular. Together with previous results, this provides simple examples of random Schrödinger operators having a singular continuous component in its spectrum.

Localization for off-diagonal disorder and for continuous Schrödinger operators

We extend the proof of localization by Delyon, Lévy, and Souillard to accommodate the Anderson model with off-diagonal disorder and the continuous Schrödinger equation with a random potential.

Metal-insulator transition in the Hartree-Fock phase diagram of the fully polarized homogeneous electron gas in two dimensions

We determine numerically the ground state of the two-dimensional fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities (