Expansion in SL_d(Z/qZ), q arbitrary

Let S be a fixed finite symmetric subset of SL_d(Z), and assume that it generates a Zariski-dense subgroup G. We show that the Cayley graphs of pi_q(G) with respect to the generating set pi_q(S) form a family of expanders, where pi_q is the projection map Z->Z/qZ

Generic Continuous Spectrum for Ergodic Schr"odinger Operators

We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page

Holder continuity of absolutely continuous spectral measures for one-frequency Schrodinger operators

We establish sharp results on the modulus of continuity of the distribution of the spectral measure for one-frequency Schrodinger operators with Diophantine frequencies in the region of absolutely continuous spectrum. More precisely, we establish 1/2-Holder continuity near almost reducible energies (an essential support of absolutely continuous spectrum). For non-perturbatively small potentials (and for the almost Mathieu operator with subcritical coupling), our results apply for all energies.Comment: 16 page

Random data Cauchy theory for supercritical wave equations II : A global existence result

We prove that the subquartic wave equation on the three dimensional ball $\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\cap_{s<1/2} H^s(\Theta)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions

On invariant tori of full dimension for 1D periodic NLS

AbstractConsider the NLS with periodic boundary conditions in 1D(0.1)iut+Δu+Mu±ɛu|u|4=0,where M is a random Fourier multiplier defined by(0.2)Mu^(n)=Vnu^(n)and (Vn)n∈Z are independently chosen in [-1,1].The quintic nonlinearity in (0.1) is unimportant and may be replaced by u|u|p-2,p∈2Z,p⩾4.We give a proof of the following fact.Theorem. For appropriate M, (0.1) has an invariant tori T (of full dimension) satisfying12e-r|n|<|qn|<2e-r|n|(n∈Z,q∈T)(r>0 is arbitrary).Remark. The statement holds in fact for most (Vn)n∈Z∈[-1,1]Z, although not explicitly proven here.Written in Fourier modes (qn)n∈Z, the Hamiltonian corresponding to (0.1) is given by(0.3)H(q,q¯)=∑(n2+Vn)|qn|2+ɛ∑n1-n2+n3-n4+n5-n6=0qn1q¯n2qn3q¯n4qn5q¯n6.The proof of Theorem 1 will proceed along the ‘usual’ KAM scheme where the perturbation is eventually removed by consecutive canonical transformations of phase space. The most relevant literature in the present context of an infinite dimensional phase space are the papers of Fröhlich et al. [Fröhlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274] and especially Pöschel [Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] on disordered systems.Both [Fröhlich, Spencer, Wayne, Localization in disordered, nonlinear dynamical systems, J. Statist. Phys. 42 (1986) 247–274, Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393] consider Hamiltonians with short-range interactions and hence these results do not apply to our problem. It turns out, however that the scheme, as elaborated on in great detail in [Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393], is still applicable to (0.3), due to special arithmetical features as will be explained in the next section. Roughly speaking, the key point is the following observation. Let (ni) be a finite set of modes, |n1|⩾|n2|⩾⋯ and(0.4)n1-n2+n3-⋯=0.In the case of a ‘near’ resonance, there is also a relation(0.5)n12-n22+n32-⋯=o(1).Unless n1=n2, one may then control |n1|+|n2| from (0.4), (0.5) by ∑j⩾3|nj|. This feature is specifically 1-dimensional and we do not know at this time how to prove a 2D-analogue of Theorem 1, considering for instance the cubic NLS iut+Δu±u|u|2=0 on T2.It should also be pointed out that almost periodic solutions on a full set of frequencies for NLS and NLW in 1D were constructed in earlier works (see [Bourgain, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, GAFA 6 (2) (1996) 201–230] and [Pöschel, On the construction of almost periodic solutions for nonlinear Schrödinger equations, Ergodic Theory Dynamical Systems 22 (5) (2002) 1537–1559]). These invariant tori (of full dimension) were obtained by successive small perturbations of finite-dimensional tori, resulting in very strong compactness properties and in fact a nonexplicit decay rate of the action variables In for n→∞. On the other hand, the construction in this paper (similarly to [Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, CMP 127 (1990) 351–393]) treats all Fourier modes at once and requires explicit and realistic decay conditions.The multiplier M=(Vn) in (0.3) is to be considered as a parameter and (0.1) a parameter-dependent equation. The role of this parameter is essential to ensure appropriate nonresonance properties of the (modulated) frequencies along the iteration. In the absence of exterior parameters, these conditions need to be realized from amplitude–frequency modulation and suitable restriction of the action-variables. This problem is harder. Indeed, a fast decay of the action-variables (enhancing convergence of the process) allows less frequency modulation and worse small divisors (cf. [Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80 (2000) 1–35])

Quasi-T\"oplitz functions in KAM theorem

We define and describe the class of Quasi-T\"oplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr\"odinger equation on the torus $T^d$, thus proving existence and stability of quasi-periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment.Comment: 34 pages, 1 figur

On Eigenvalue spacings for the 1-D Anderson model with singular site distribution

We study eigenvalue spacings and local eigenvalue statistics for 1D lattice Schrodinger operators with Holder regular potential, obtaining a version of Minami's inequality and Poisson statistics for the local eigenvalue spacings. The main additional new input are regular properties of the Furstenberg measures and the density of states obtained in some of the author's earlier work.Comment: 13 page