4,345 research outputs found
Quantum Evolution and Anticipation
In a previous paper we have investigated quantum states evolving into
mutually orthogonal states at equidistant times, and the quantum anticipation
effect exhibited by measurements at one half step. Here we extend our analyzes
of quantum anticipation to general type quantum evolutions and spectral
measures and prove that quantum evolutions possessing an embedded orthogonal
evolution are characterized by positive joint spectral measure. Furthermore, we
categorize quantum evolution, assess anticipation strength and provide a
framework of analytic tools and results, thus preparing for further
investigation and experimental verification of anticipation in concrete
physical situations such as the H-atom, which we have found to exhibit
anticipation.Comment: 22 page
Random weighted Sobolev inequalities and application to quantum ergodicity
This paper is a continuation of Poiret-Robert-Thomann (2013) where we studied
a randomisation method based on the Laplacian with harmonic potential. Here we
extend our previous results to the case of any polynomial and confining
potential on . We construct measures, under concentration
type assumptions, on the support of which we prove optimal weighted Sobolev
estimates on . This construction relies on accurate estimates on
the spectral function in a non-compact configuration space. Then we prove
random quantum ergodicity results without specific assumption on the classical
dynamics. Finally, we prove that almost all basis of Hermite functions is
quantum uniquely ergodic.Comment: Clarifications added in the part concerning QU
On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential
We prove a global existence result with initial data of low regularity, and
prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system
with small non linear term but with a possibly large exterior confining
potential in dimension and . The proof relies on a fixed point
argument using sharp estimates (at short and long time scales) of the
semi-group associated to the Fokker-Planck operator, which were obtained by the
first author.Comment: 29 pages. To appear in Journal of Functional Analysi
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