42 research outputs found
Regularity for eigenfunctions of Schr\"odinger operators
We prove a regularity result in weighted Sobolev spaces (or
Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator.
More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space
obtained by blowing up the set of singular points of the Coulomb type potential
V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N}
\frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u
in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution
sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0.
Our result extends to the case when b_j and c_{ij} are suitable bounded
functions on the blown-up space. In the single-electron, multi-nuclei case, we
obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy
Spectral problems in open quantum chaos
This review article will present some recent results and methods in the study
of 1-particle quantum or wave scattering systems, in the semiclassical/high
frequency limit, in cases where the corresponding classical/ray dynamics is
chaotic. We will focus on the distribution of quantum resonances, and the
structure of the corresponding metastable states. Our study includes the toy
model of open quantum maps, as well as the recent quantum monodromy operator
method.Comment: Compared with the previous version, misprints and typos have been
corrected, and the bibliography update
Local smoothing for scattering manifolds with hyperbolic trapped sets
We prove a resolvent estimate for the Laplace-Beltrami operator on a
scattering manifold with a hyperbolic trapped set, and as a corollary deduce
local smoothing. We use a result of Nonnenmacher-Zworski to provide an estimate
near the trapped region, a result of Burq and Cardoso-Vodev to provide an
estimate near infinity, and the microlocal calculus on scattering manifolds to
combine the two.Comment: 16 pages. Published version available at
http://www.springerlink.com/content/r663321331243288/?p=5ad2fe4778a742e4949de2030a409358&pi=1
Local energy decay for several evolution equations on asymptotically Euclidean manifolds
International audienc
Asymptotically flat Einstein-Maxwell fields are inheriting
We prove that Maxwell fields of asymptotically flat solutions of the Einstein–Maxwell equations inherit the stationarity of the metric