The Canonical Function Method and its applications in Quantum Physics

The Canonical Function Method (CFM) is a powerful method that solves the radial Schr\"{o}dinger equation for the eigenvalues directly without having to evaluate the eigenfunctions. It is applied to various quantum mechanical problems in Atomic and Molecular physics with presence of regular or singular potentials. It has also been developed to handle single and multiple channel scattering problems where the phaseshift is required for the evaluation of the scattering cross-section. Its controllable accuracy makes it a valuable tool for the evaluation of vibrational levels of cold molecules, a sensitive test of Bohr correspondance principle and a powerful method to tackle local and non-local spin dependent problems.Comment: 30 pages, 12 figures- To submit to Reviews of Modern Physic

General treatment of isocurvature perturbations and non-Gaussianities

We present a general formalism that provides a systematic computation of the linear and non-linear perturbations for an arbitrary number of cosmological fluids in the early Universe going through various transitions, in particular the decay of some species (such as a curvaton or a modulus). Using this formalism, we revisit the question of isocurvature non-Gaussianities in the mixed inflaton-curvaton scenario and show that one can obtain significant non-Gaussianities dominated by the isocurvature mode while satisfying the present constraints on the isocurvature contribution in the observed power spectrum. We also study two-curvaton scenarios, taking into account the production of dark matter, and investigate in which cases significant non-Gaussianities can be produced.Comment: Substantial improvements with respect to the first version. In particular, we added a discussion on the confrontation of the models with future observational data. This version is accepted for publication in JCA

Massive scalar states localized on a de Sitter brane

We consider a brane scenario with a massive scalar field in the five-dimensional bulk. We study the scalar states that are localized on the brane, which is assumed to be de Sitter. These localized scalar modes are massive in general, their effective four-dimensional mass depending on the mass of the five-dimensional scalar field, on the Hubble parameter in the brane and on the coupling between the brane tension and the bulk scalar field. We then introduce a purely four-dimensional approach based on an effective potential for the projection of the scalar field in the brane, and discuss its regime of validity. Finally, we explore the quasi-localized scalar states, which have a non-zero width that quantifies their probability of tunneling from the brane into the bulk.Comment: 14 pages; 5 figure

On the classification of normal G-varieties with spherical orbits

In this article, we investigate the geometry of reductive group actions on algebraic varieties. Given a connected reductive group $G$, we elaborate on a geometric and combinatorial approach based on Luna-Vust theory to describe every normal $G$-variety with spherical orbits. This description encompasses the classical case of spherical varieties and the theory of $\mathbb{T}$-varieties recently introduced by Altmann, Hausen, and S\"uss

Decomposition theorem and torus actions of complexity one

We algorithmically compute the intersection cohomology Betti numbers of any complete normal algebraic variety with a torus action of complexity one

Quantization of scalar perturbations in brane-world inflation

We consider a quantization of scalar perturbations about a de Sitter brane in a 5-dimensional anti-de Sitter (AdS) bulk spacetime. We first derive the second order action for a master variable $\Omega$ for 5-dimensional gravitational perturbations. For a vacuum brane, there is a continuum of normalizable Kaluza-Klein (KK) modes with $m>3H/2$. There is also a light radion mode with $m=\sqrt{2}H$ which satisfies the junction conditions for two branes, but is non-normalizable for a single brane model. We perform the quantization of these bulk perturbations and calculate the effective energy density of the projected Weyl tensor on the barne. If there is a test scalar field perturbation on the brane, the $m^2 = 2H^2$ mode together with the zero-mode and an infinite ladder of discrete tachyonic modes become normalizable in a single brane model. This infinite ladder of discrete modes as well as the continuum of KK modes with $m>3H/2$ introduce corrections to the scalar field perturbations at first-order in a slow-roll expansion. We derive the second order action for the Mukhanov-Sasaki variable coupled to the bulk perturbations which is needed to perform the quantization and determine the amplitude of scalar perturbations generated during inflation on the brane.Comment: 14 page

On intersection cohomology with torus actions of complexity one

The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus T, one of our result determines the intersection cohomology Betti numbers of any normal projective T-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context