Limiting absorption principle and perfectly matched layer method for Dirichlet Laplacians in quasi-cylindrical domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.Comment: to appear in SIAM Journal on Mathematical Analysi

Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear Fokker-Planck equation) coupled with Navier-Stokes equations in two dimensions. The proof uses a deteriorating regularity estimate and the tensorial structure of the main nonlinear terms

Wave decay on convex co-compact hyperbolic manifolds

For convex co-compact hyperbolic quotients X=\Gamma\backslash\hh^{n+1}, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the Hausdorff dimension $\delta$ of the limit set is less than $n/2$, then u(t) = C_\delta(f) e^{(\delta-\ndemi)t} / \Gamma(\delta-n/2+1) + e^{(\delta-\ndemi)t} R(t) where $C_{\delta}(f)\in C^\infty(X)$ and ||R(t)||=\mc{O}(t^{-\infty}). We explain, in terms of conformal theory of the conformal infinity of $X$, the special cases \delta\in n/2-\nn where the leading asymptotic term vanishes. In a second part, we show for all \eps>0 the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip \{-n\delta-\eps<\Re(\la)<\delta\}. As a byproduct we obtain a lower bound on the remainder $R(t)$ for generic initial data $f$.Comment: 18 page

Scattering Theory Approach to Random Schroedinger Operators in One Dimension

Methods from scattering theory are introduced to analyze random Schroedinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynamic limit" of the spectral shift function per unit length of the interaction region. This density is shown to be equal to the difference of the densities of states for the free and the interacting Hamiltonians. Based on this construction, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggest a way how to extend this notion to the higher dimensional case. This prescription also allows a characterization of those energies which have vanishing Lyapunov exponent.Comment: 1 figur

Note on Global Regularity for 2D Oldroyd-B Fluids with Diffusive Stress

We prove global regularity of solutions of Oldroyd-B equations in 2 spatial dimensions with spatial diffusion of the polymeric stresses

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Surfactant effect in heteroepitaxial growth. The Pb - Co/Cu(111) case

A MonteCarlo simulations study has been performed in order to study the effect of Pb as surfactant on the initial growth stage of Co/Cu(111). The main characteristics of Co growing over Cu(111) face, i.e. the decorated double layer steps, the multiple layer islands and the pools of vacancies, disappear with the pre-evaporation of a Pb monolayer. Through MC simulations, a full picture of these complex processes is obtained. Co quickly diffuses through the Pb monolayer exchanging place with Cu atoms at the substrate. The exchange process diffusion inhibits the formation of pure Co islands, reducing the surface stress and then the formation of multilayer islands and the pools of vacancies. On the other hand, the random exchange also suppress the nucleation preferential sites generated by Co atoms at Cu steps, responsible of the step decoration.Comment: 4 pages, latex, 2 figures embedded in the tex

Distribution of resonances for open quantum maps

We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.Comment: 60 pages, no figures (numerical results are shown in other references

Reflectivity Anisotropy Spectra of Cu- and Ag- (110) surfaces from {\it ab initio} theory

We are able to disentagle the effects of the intraband and interband parts of the bulk dielectric function on the bare dielectric anisotropy of the surface. We show how the position, sign and amplitude of the structures observed in such spectra depend on the above quantities. The lineshape of all the calculated structures agree very well with the ones observed experimentally for samples treated by suitable surface cleaning. In particular, we reproduce the observed single peak structure of Ag at high energy, found to represent a state of the clean surface different from the one giving the originally observed double peak structure. This results is not reproduced by the 'local field' model.Comment: 4 pages, 3 figures. submitted to Phys. Rev. Let