12,110 research outputs found
Semi-classical spectral estimates for Schr\"odinger operators at a critical level. Case of a degenerate maximum of the potential
We study the semi-classical trace formula at a critical energy level for a
Schr\"odinger operator on . We assume here that the potential
has a totally degenerate critical point associated to a local maximum. The main
result, which establishes the contribution of the associated equilibrium in the
trace formula, is valid for all time in a compact subset of and
includes the singularity in . For these new contributions the asymptotic
expansion involves the logarithm of the parameter . Depending on an explicit
arithmetic condition on the dimension and the order of the critical point, this
logarithmic contribution can appear in the leading term.Comment: 27 pages, perhaps to be revise
Spectral estimates for degenerate critical levels
We establish spectral estimates at a critical energy level for -pseudors .
Via a trace formula, we compute the contribution of isolated (non-extremum)
critical points under a condition of "real principal type". The main result
holds for all dimensions, for a singularity of any finite order and can be
invariantly expressed in term of the geometry of the singularity. When the
singularities are not integrable on the energy surface the results are
significative since the order w.r.t. of the spectral distributions are
bigger than in the regular setting.Comment: 24 page
Asymptotic approximation of degenerate fiber integrals
We study asymptotics of fiber integrals depending on a large parameter. When
the critical fiber is singular, full-asymptotic expansions are established in
two different cases : local extremum and isolated real principal type
singularities. The main coefficients are computed and invariantly expressed. In
the most singular cases it is shown that the leading term of the expansion is
related to invariant measures on the spherical blow-up of the singularity. The
results can be applied to certain degenerate oscillatory integrals which occur
in spectral analysis and quantum mechanics.Comment: 22 pages, perhaps to be revise
Quantum technology: single-photon source
This report is a synthesis of my master thesis internship at the National
Institute of Informatics (NII) in Tokyo, Japan, that lasted during the summer
of year 2012. I worked in the Quantum Information Science Theory (QIST) group
under supervision of Prof. Kae Nemoto and Dr. Simon Devitt. This group works on
theoretical and experimental implementations of quantum information science.
The aim of my project was to study and improve quantum optical systems. I first
studied different fields and systems of quantum information science. Then I
focused my research on single-photon sources, entangled photon sources and
interferometric photonic switches. Finally, I found some strategies to design
an efficient and optimized single-photon source that could be built with
today's technologies. This report describes in details the created and
optimized design of a single-photon source based on time and space multiplexing
of Spontaneous Parametric Downconversion (SPDC) sources.Comment: Research extract of Master thesis report. Defended in September 2012.
Declassified by the NII in February 201
Fundamental solutions for a class of non-elliptic homogeneous differential operators
We compute temperate fundamental solutions of homogeneous differential
operators with real-principal type symbols. Via analytic continuation of
meromorphic distributions, fundamental solutions for these non-elliptic
operators can be constructed in terms of radial averages and invariant
distributions on the unit sphere.Comment: 15 pages, perhaps to be revise
Equilibrium and eigenfunctions estimates in the semi-classical regime
We establish eigenfunctions estimates, in the semi-classical regime, for
critical energy levels associated to an isolated singularity. For Schr\"odinger
operators, the asymptotic repartition of eigenvectors is the same as in the
regular case, excepted in dimension 1 where a concentration at the critical
point occurs. This principle extends to pseudo-differential operators and the
limit measure is the Liouville measure as long as the singularity remains
integrable.Comment: 13 pages, 1 figure, perhaps to be revise
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