High Frequency Resolvent Estimates for Perturbations by Large Long-range Magnetic Potentials and Applications to Dispersive Estimates

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form ${G=\left(i\nabla+b(x)\right)^2+V(x)}$ on ${L^2({\bf R}^n), n\ge 3}$ , where the magnetic potential b(x) and the electric potential V(x) are long-range and large. As an application, we prove dispersive estimates for the wave group ${{\rm e}^{it\sqrt{G}}}$ in the case n = 3 for potentials b(x), V(x) = O(|x|^?2-δ ) for ${|x|\gg 1}$ , where δ > 0.     

Effects of the agglomeration state on the Raman properties of Co3O4 nanoparticles

The features of the Raman spectra of Co₃O₄ 30‐nm nanoparticles depend strongly on their agglomeration state. When measured at low incident laser power, the spectrum of isolated nanoparticles corresponds to that found in bulk materials, whereas the agglomerated nanoparticles present a clear red‐shift and broadening of the Raman bands. On the other hand, when measured at even lower power, both agglomerated and isolated nanoparticles show the same spectrum of microscopic particles. These effects have been studied by variations of the 532‐nm laser power and the environmental temperature. The thermal dependence of Raman spectra of agglomerated nanoparticles is different to that of isolated nanoparticles but is comparable to the one of bulk material. The different behaviour of the nanoparticles at different agglomeration state is associated to the transmission of phonons among the particles. On the other hand, an increase of the laser power causes a larger number of acoustic phonons, producing a variation of the vibration anharmonicity of the nanoparticles. This increase is more pronounced in the agglomerated nanoparticles, due to the transmission of phonons, causing a much intense modification of the Raman spectrum produced by the laser power. These results clearly indicate that the agglomeration state of the nanoparticles affects their Raman properties. Copyright © 2012 John Wiley & Sons, Ltd.     

Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

Discrete moving breather collisions in a Klein-Gordon chain of oscillators

We study the collisions of moving breathers with the same frequency, traveling with opposite directions within a Klein-Gordon chain of oscillators. Two types of collisions have been analyzed: symmetric and non-symmetric, head-on collisions. For low enough frequency the outcome is strongly dependent of the dynamical states of the two colliding breathers just before the collision. For symmetric collisions, several results can be observed: breather generation, with the formation of a trapped breather and two new moving breathers; breather reflection; generation of two new moving breathers; and breather fusion bringing about a trapped breather. For non-symmetric collisions some possible results are: breather generation, with the formation of three new moving breathers; breather fusion, originating a new moving breather; breather trapping with breather reflection; generation of two new moving breathers; and two new moving breathers traveling as a bound state. Breather annihilation has never been observed.     

Discrete problems associated to elliptic equations

This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 John Wiley & Sons, Ltd.     

A new model of quantum chaotic billiards: application to granular metals

We discuss the properties of a recently proposed model of quantum chaotic billiards in two and three dimensions. The model is based on a tight-binding Hamiltonian in which the energies of the atomic levels at the boundary sites are chosen at random between -W/2 and W/2. The energy spectra show a complex behavior with regions that obey Wigner-Dyson statistics, and regions with localized and quasi-ideal states distributed according to Poisson statistics. Whereas at low energies long-range energy fluctuations follow Random Matrix Theory (RMT) for all W, at high energies fluctuations are below (above) RMT for small (large) W. For small W, the mean free path l is proportional to L/W ², L being the system size, and reaches a minimum for W of the order of the band width, at which l ≈ L/2. In 3D we found that the energy fluctuations of the highest occupied level are much larger than the average interlevel spacing. This provides an explanation for autoionization effects of the grains in granular metals.