Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPropagation des singularites et resonances
75
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the framework of semiclassical resonances, we make more precise the link between polynomial estimates of the extension of the resolvent and propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations.
rate research
Read More
We study semiclassical resonances generated by homoclinic trapped sets. First, under some general assumptions, we prove that there is no resonance in a region below the real axis. Then, we obtain a quantization rule and the asymptotic expansion of the resonances when there is a finite number of homoclinic trajectories. The same kind of results is proved for homoclinic sets of maximal dimension. Next, we generalize to the case of homoclinic/heteroclinic trajectories and we study the three bump case. In all these settings, the resonances may either accumulate on curves or form clouds. We also describe the corresponding resonant states.
We give the semiclassical asymptotic of barrier-top resonances for Schr{o}dinger operators on ${mathbb R}^{n}$, $n geq 1$, whose potential is $C^{infty}$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained. Our proof is based on a propagation of singularities theorem at a hyperbolic fixed point that we establish here. This last result refines a theorem of the same authors, and its proof follows another approach.
We investigate plasmon resonances for curved nanorods which present anisotropic geometries. We analyze quantitative properties of the plasmon resonance and its relationship to the metamaterial configurations and the anisotropic geometries of the nanorods. Based on delicate and subtle asymptotic and spectral analysis of the layer potential operators, particularly the Neumann-Poincare operators, associated with anisotropic geometries, we derive sharp asymptotic formulae of the corresponding scattering field in the quasi-static regime. By carefully analyzing the asymptotic formulae, we establish sharp conditions that can ensure the occurrence of the plasmonic resonance. The resonance conditions couple the metamaterial parameters, the wave frequency and the nanorod geometry in an intricate but elegant manner. We provide thorough resonance analysis by studying the wave fields both inside and outside the nanorod. Furthermore, our quantitative analysis indicates that different parts of the nanorod induce varying degrees of resonance. Specifically, the resonant strength at the two end-parts of the curved nanorod is more outstanding than that of the facade-part of the nanorod. This paper presents the first theoretical study on plasmon resonances for nanostructures within anisotropic geometries.
Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension n+1 with an isolated singularity at 0. Let (f_R,0) : (X_R, 0) --> (R,0) a real analytic germ with an isolated singularity at 0, such that its complexification f_C vanishes on the singular set S of X_C. We also assume that X_R-[0] is orientable. To each $ A in H^{0}(X_{mathbb{R}} - lbrace 0 rbrace ,mathbb {C}) $ we associate a $n-$cycle $ Gamma(A) $ (explicitly described) in the complex Milnor fiber of $f_{mathbb{C}}$ at 0 such that the non trivial terms in the asymptotic expansions of the oscillating integrals $ int_{A} e^{itau f(x)} phi(x) $ when $ tau to pm infty $ can be read from the spectral decomposition of $Gamma(A) $ relative to the monodromy of $f_{mathbb{C}}$ at 0 .
We prove that for the Martinet wave equation with flat metric, which a subelliptic wave equation, singularities can propagate at any speed between 0 and 1 along any singular geodesic. This is in strong contrast with the usual propagation of singularities at speed 1 for wave equations with elliptic Laplacian.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
