30 research outputs found
Delocalization of quasimodes on the disk
This note deals with semiclassical measures associated to {(sufficiently
accurate)} quasimodes for the Laplace-Dirichlet operator on the disk.
In this time-independent set-up, we simplify the statements of our preprint
arXiv:1406.0681 and their proofs. We describe the restriction of semiclassical
measures to every invariant torus in terms of two-microlocal measures. As
corollaries, we show regularity and delocalization properties for limit
measures of : these are absolutely continuous in the interior of
the disk and charge every open set intersecting the boundary.Comment: arXiv admin note: text overlap with arXiv:1406.068
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