Semiclassical quantization of the hydrogen atom in crossed electric and magnetic fields

The S-matrix theory formulation of closed-orbit theory recently proposed by Granger and Greene is extended to atoms in crossed electric and magnetic fields. We then present a semiclassical quantization of the hydrogen atom in crossed fields, which succeeds in resolving individual lines in the spectrum, but is restricted to the strongest lines of each n-manifold. By means of a detailed semiclassical analysis of the quantum spectrum, we demonstrate that it is the abundance of bifurcations of closed orbits that precludes the resolution of finer details. They necessitate the inclusion of uniform semiclassical approximations into the quantization process. Uniform approximations for the generic types of closed-orbit bifurcation are derived, and a general method for including them in a high-resolution semiclassical quantization is devised

Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case

We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, -\la_1(\om)0 and $\beta\neq 0$, where \lambda_1(\om) is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution (u_\bb, v_\bb) {\it for any $\beta\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of (u_\bb, v_\bb) as $\beta\to -\infty$ and phase separation is expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case \la_1=\la_2, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP

Bubble concentration on spheres for supercritical elliptic problems

We consider the supercritical Lane-Emden problem (P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} where $\mathcal A$ is an annulus in \rr^{2m}, $m\ge2$ and p_\eps={(m+1)+2\over(m+1)-2}-\eps, \eps>0. We prove the existence of positive and sign changing solutions of (P_\eps) concentrating and blowing-up, as \eps\to0, on $(m-1)-$dimensional spheres. Using a reduction method (see Ruf-Srikanth (2010) J. Eur. Math. Soc. and Pacella-Srikanth (2012) arXiv:1210.0782)we transform problem (P_\eps) into a nonhomogeneous problem in an annulus \mathcal D\subset \rr^{m+1} which can be solved by a Ljapunov-Schmidt finite dimensional reduction

The hydrogen atom in an electric field: Closed-orbit theory with bifurcating orbits

Closed-orbit theory provides a general approach to the semiclassical description of photo-absorption spectra of arbitrary atoms in external fields, the simplest of which is the hydrogen atom in an electric field. Yet, despite its apparent simplicity, a semiclassical quantization of this system by means of closed-orbit theory has not been achieved so far. It is the aim of this paper to close that gap. We first present a detailed analytic study of the closed classical orbits and their bifurcations. We then derive a simple form of the uniform semiclassical approximation for the bifurcations that is suitable for an inclusion into a closed-orbit summation. By means of a generalized version of the semiclassical quantization by harmonic inversion, we succeed in calculating high-quality semiclassical spectra for the hydrogen atom in an electric field

Dvoretzky type theorems for multivariate polynomials and sections of convex bodies

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ of the polynomial field conjecture

Evolution of Surface Morphology of Thermo-Mechanically Cycled NiCoCrAlY Bond Coats

We investigate morphological surface instabilities on bond coat surfaces of thermal barrier coatings, induced due to thermo-mechanical loading. Experimental results of hollow circular cylindrical specimens, consisting of a directionally solidified superalloy (IN 100 DS) coated with a NiCoCrAlY bond coat, show that the morphological instabilities are strongly dependent on the load conditions. In particular, the morphological instabilities develop during thermal cycling with a thermal gradient over the cylinder wall, whereas the surface remains smooth for thermal cyclic conditions without a gradient. Furthermore, if a cyclic, axial tensile force is applied (synchronized with the thermal cycling), the morphological instabilities become aligned with the axial direction. We discuss a model, quantified by finite element simulations, capturing this behavior and elucidating the thermo-mechanical response

Evolution of Surface Morphology of Thermo-Mechanically Cycled NiCoCrAlY Bond Coats

We investigate morphological surface instabilities on bond coat surfaces of thermal barrier coatings, induced due to thermo-mechanical loading. Experimental results of hollow circular cylindrical specimens, consisting of a directionally solidified superalloy (IN 100 DS) coated with a NiCoCrAlY bond coat, show that the morphological instabilities are strongly dependent on the load conditions. In particular, the morphological instabilities develop during thermal cycling with a thermal gradient over the cylinder wall, whereas the surface remains smooth for thermal cyclic conditions without a gradient. Furthermore, if a cyclic, axial tensile force is applied (synchronized with the thermal cycling), the morphological instabilities become aligned with the axial direction. We discuss a model, quantified by finite element simulations, capturing this behavior and elucidating the thermo-mechanical response

Secondary metabolites of the argan tree (Morocco) may have disease prevention properties

The argan tree (Argania spinosa L. Skeels) is native to Morocco, where after the Holly oak it constitutes the second most common tree in the country. Recent studies suggest that dietary argan oil, an endemic seed oil from argan fruits, may have a relevant role in disease prevention, and its consumption could protect against atherosclerosis and cancer. Unfortunately, in less than a century, more than a third of the forest has disappeared. It is therefore imperative to improve the tree\'s production potential so that it can regain its key position in the agricultural systems of the region. On the basis of ethnobotanical knowledge, researchers are screening metabolites of this rare plant to identify bioactive compounds for the development of new therapeutic agents and food supplements. This includes studies on secondary metabolites with chemopreventive activities. In this review, a complete outline of components (triglycerides, unsaponifiable, phenolic antioxidants and aroma constituents) are described. Finally, a discussion of the biological functions of the polar and non-polar A. spinosa products which have been evaluated using a range of in vitro bioassays are described.African Journal of Biotechnology Vol. 4 (5), pp. 381-388, 200

Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits

With increasing energy the diamagnetic hydrogen atom undergoes a transition from regular to chaotic classical dynamics, and the closed orbits pass through various cascades of bifurcations. Closed orbit theory allows for the semiclassical calculation of photoabsorption spectra of the diamagnetic hydrogen atom. However, at the bifurcations the closed orbit contributions diverge. The singularities can be removed with the help of uniform semiclassical approximations which are constructed over a wide energy range for different types of codimension one and two catastrophes. Using the uniform approximations and applying the high-resolution harmonic inversion method we calculate fully resolved semiclassical photoabsorption spectra, i.e., individual eigenenergies and transition matrix elements at laboratory magnetic field strengths, and compare them with the results of exact quantum calculations.Comment: 26 pages, 9 figures, submitted to J. Phys.

The significance of new records of benthic red algae (Rhodophyta) for Hainan Island (and China) between 1990 and 2016

We present an annotated list of new finds of red algae from Hainan Island, Southern China, including those found in 1990 and 1992 during the German-Chinese expeditions to Hainan Island and in 2008–2016 by Titlyanova, Titlyanov, and Li. Between 1990 and 1992, a total of 64 taxa of red algae were newly recorded for Hainan Island. Of these 15 species were new records for China. During the period 2008–2016, a further 54 taxa were newly recorded for Hainan Island, of which 20 were new records for China. The full list of new taxa includes taxonomic forms, dates, and locales, together with known biogeographical distributions. During both periods, the apparent enrichment of red algal marine flora has occurred in a similar way—mainly at the expense of epiphytes with filamentous, thin-filamentous, and finely branched forms. We believe that the changes in the flora of Hainan Island have been influenced by both anthropogenic and natural factors including in particular exploitation of herbivores, nutrient pollution, and coral bleaching