122 research outputs found
Wigner Measure Propagation and Conical Singularity for General Initial Data
We study the evolution of Wigner measures of a family of solutions of a
Schr\"odinger equation with a scalar potential displaying a conical
singularity. Under a genericity assumption, classical trajectories exist and
are unique, thus the question of the propagation of Wigner measures along these
trajectories becomes relevant. We prove the propagation for general initial
data.Comment: 24 pages, 1 figur
Defect measures on graded lie groups
In this article, we define a generalisation of microlocal defect measures
(also known as H-measures) to the setting of graded nilpotent Lie groups. This
requires to develop the notions of homogeneous symbols and classical
pseudo-differential calculus adapted to this setting and defined via the
representations of the groups. Our method relies on the study of the C
*-algebra of 0-homogeneous symbols. Then, we compute microlocal defect measures
for concentrating and oscillating sequences, which also requires to investigate
the notion of oscillating sequences in graded Lie groups. Finally, we discuss
compacity compactness approaches in the context of graded nilpotent Lie groups
Analysis of the Energy Decay of a Degenerated Thermoelasticity System
In this paper, we study a system of thermoelasticity with a degenerated
second order operator in the Heat equation. We analyze the evolution of the
energy density of a family of solutions. We consider two cases: when the set of
points where the ellipticity of the Heat operator fails is included in a
hypersurface and when it is an open set. In the first case and under special
assumptions, we prove that the evolution of the energy density is the one of a
damped wave equation: propagation along the rays of geometric optic and damping
according to a microlocal process. In the second case, we show that the energy
density propagates along rays which are distortions of the rays of geometric
optic.Comment: 28 page
Dispersive estimates for the Schr\"odinger operator on step 2 stratified Lie groups
The present paper is dedicated to the proof of dispersive estimates on
stratified Lie groups of step 2, for the linear Schr\"odinger equation
involving a sublaplacian. It turns out that the propagator behaves like a wave
operator on a space of the same dimension p as the center of the group, and
like a Schr\"odinger operator on a space of the same dimension k as the radical
of the canonical skew-symmetric form, which suggests a decay with exponant
-(k+p-1)/2. In this article, we identify a property of the canonical
skew-symmetric form under which we establish optimal dispersive estimates with
this rate. The relevance of this property is discussed through several
examples
Semiclassical Completely Integrable Systems : Long-Time Dynamics And Observability Via Two-Microlocal Wigner Measures
We look at the long-time behaviour of solutions to a semi-classical
Schr\"odinger equation on the torus. We consider time scales which go to
infinity when the semi-classical parameter goes to zero and we associate with
each time-scale the set of semi-classical measures associated with all possible
choices of initial data. On each classical invariant torus, the structure of
semi-classical measures is described in terms of two-microlocal measures,
obeying explicit propagation laws. We apply this construction in two
directions. We first analyse the regularity of semi-classical measures, and we
emphasize the existence of a threshold : for time-scales below this threshold,
the set of semi-classical measures contains measures which are singular with
respect to Lebesgue measure in the "position" variable, while at (and beyond)
the threshold, all the semi-classical measures are absolutely continuous in the
"position" variable, reflecting the dispersive properties of the equation.
Second, the techniques of two- microlocal analysis introduced in the paper are
used to prove semiclassical observability estimates. The results apply as well
to general quantum completely integrable systems.Comment: This article contains and develops the results of hal-00765928. arXiv
admin note: substantial text overlap with arXiv:1211.151
Long-time dynamics of completely integrable Schr\"odinger flows on the torus
In this article, we are concerned with long-time behaviour of solutions to a
semi-classical Schr\"odinger-type equation on the torus. We consider time
scales which go to infinity when the semi-classical parameter goes to zero and
we associate with each time-scale the set of semi-classical measures associated
with all possible choices of initial data. We emphasize the existence of a
threshold: for time-scales below this threshold, the set of semi-classical
measures contains measures which are singular with respect to Lebesgue measure
in the "position" variable, while at (and beyond) the threshold, all the
semi-classical measures are absolutely continuous in the "position" variable.Comment: 41 page
Wigner measures and codimension two crossings
This article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels. Codimension two crossings are the simplest energy level crossings, which affect the Born–Oppenheimer approximation in the zeroth order term. The model we study is a two-level Schrödinger equation with a Laplacian as kinetic operator and a matrix-valued linear potential, whose eigenvalues cross, if the two nucleonic coordinates equal zero. We discuss the case of well-localized initial data and obtain a description of the wavefunction’s two-scaled Wigner measure and of the weak limit of its position density, which is valid globally in time
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