Asymptotic integration of nonlinear systems of differential equations whose phase portrait is foliated on invariant tori

We consider the class of autonomous systems $\dot x=f(x)$, where $x \in {\bf R}^{2n}$, $f \in C^1({\bf R}^{2n})$ whose phase portrait is a Cartesian product of $n$ two-dimensional {\em centres}. We also consider perturbations of this system, namely $\dot x=f(x)+g(t,x)$, where $g \in C^1({\bf R}\times{\bf R}^{2n})$ and $g$ is asymptotically small, that is $g\Rightarrow 0$ as $t\to +\infty$ uniformly with respect to $x$. The rate of decrease of $g$ is assumed to be $t^{-p}$ where $p>1$. We prove under this conditions the existence of bounded solutions of the perturbed system and discuss their convergence to solutions of the unperturbed system. This convergence depends on $p$. Moreover, we show that the original unperturbed system may be reduced to the form $\dot r=0$, $\dot\theta=A(r)$, and taking $r\in {\bf R}^m_{+}$, $\theta\in {\bf T}^n$, where ${\bf T}^n$ denotes the $n$-dimensional torus, we investigate the more general case of systems whose phase portrait is foliated on invariant tori. We notice that integrable Hamiltonian systems are of the same nature. We give also several examples, showing that the conditions of our theorems cannot be improved

Lieb-Thirring inequalities on some manifolds

We prove Lieb-Thirring inequalities with improved constants on the two-dimensional sphere and the two-dimensional torus. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues

Stoponium Search at Photon Linear Collider

In some supersymmetric extensions of the Standard Model fairly light superpartner of t-quark is predicted, which may form bound states ({\it stoponiums}) under certain conditions. We study prospects of search for stoponium at the future Photon Linear Collider. It is found that this machine could be the best machine for discovery and study of these resonances at some scenarios of supersymmetric extension of the Standard Model. In particular, if the $hh$ decay channel is dominant stoponium could be observed at the beginning of PLC run with collision energy tuned at the stoponium mass. If this channel is kinematically closed stoponium could be discovered in $gg$, $\gamma\gamma$ and $ZZ$ decay channels but higher statistics are needed. Effects of the stoponium-Higgs mixing and degeneracy are briefly discussed.Comment: 11 pages, 2 figures added, corrections taken into account result in increasing of signal significanc

Generalized binomial distribution in photon statistics

The photon-number distribution between two parts of a given volume is found for an arbitrary photon statistics. This problem is related to the interaction of a light beam with a macroscopic device, for example a diaphragm, that separates the photon flux into two parts with known probabilities. To solve this problem, a Generalized Binomial Distribution (GBD) is derived that is applicable to an arbitrary photon statistics satisfying probability convolution equations. It is shown that if photons obey Poisson statistics then the GBD is reduced to the ordinary binomial distribution, whereas in the case of Bose-Einstein statistics the GBD is reduced to the Polya distribution. In this case, the photon spatial distribution depends on the phase-space volume occupied by the photons. This result involves a photon bunching effect, or collective behavior of photons that sharply differs from the behavior of classical particles. It is shown that the photon bunching effect looks similar to the quantum interference effect.Comment: 8 pages, 4 figure