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 $\mathbb{R}^{n}$. 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 $\mathbb{R}$ and includes the singularity in $t=0$. For these new contributions the asymptotic expansion involves the logarithm of the parameter $h$. 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

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

Spectral estimates for degenerate critical levels

We establish spectral estimates at a critical energy level for $h$-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. $h$ of the spectral distributions are bigger than in the regular setting.Comment: 24 page

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