From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables

This article is a part of a project investigating the relationship between the dynamics of completely integrable or close to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for families of Laplace-Beltrami operators with Dirichlet, Neumann or Robin boundary conditions, associated with C^1 families of billiard tables. We introduce a notion of weak isospectrality for such deformations. The main dynamical assumption on the initial billiard table is that the corresponding billiard ball map or an iterate of it has a Kronecker invariant torus with a Diophantine frequency and that the corresponding Birkhoff Normal Form is nondegenerate in Kolmogorov sense. Then we obtain C^1 families of Kronecker tori with Diophantine frequencies. If the family of the Laplace-Beltrami operators satisfies the weak isospectral condition, we prove that the average action on the tori and the Birkhoff Normal Form of the billiard ball maps remain the same along the perturbation. As an application we obtain infinitesimal spectral rigidity for Liouville billiard tables in dimensions two and three. Applications are obtained also for strictly convex billiard tables of dimension two as well as in the case when the initial billiard table admits an elliptic periodic billiard trajectory. Spectral rigidity of billard tables close elliptical billiard tables is obtained. The results are based on a construction of C^1 families of quasi-modes associated with the Kronecker tori and on suitable KAM theorems for C^1 families of Hamiltonians.Comment: 170 pages; new results about the spectral rigidity of elliptical billiard tables; new Modified Iterative Lemma in the proof of KAM theorem with parameter

Trapped fermions with density imbalance in the BEC limit

We analyze the effects of imbalancing the populations of two-component trapped fermions, in the BEC limit of the attractive interaction between different fermions. Starting from the gap equation with two fermionic chemical potentials, we derive a set of coupled equations that describe composite bosons and excess fermions. We include in these equations the processes leading to the correct dimer-dimer and dimer-fermion scattering lengths. The coupled equations are then solved in the Thomas-Fermi approximation to obtain the density profiles for composite bosons and excess fermions, which are relevant to the recent experiments with trapped fermionic atomsComment: 5 pages, 4 figure

Electron-Positron Pair Production in Space- or Time-Dependent Electric Fields

Treating the production of electron and positron pairs by a strong electric field from the vacuum as a quantum tunneling process we derive, in semiclassical approximation, a general expression for the pair production rate in a $z$-dependent electric field $E(z)$ pointing in the $z$-direction. We also allow for a smoothly varying magnetic field parallel to $E(z)$. The result is applied to a confined field $E(z)\not=0$ for $|z|\lesssim \ell$, a semi-confined field $E(z)\not=0$ for $z\gtrsim 0$, and a linearly increasing field $E(z)\sim z$. The boundary effects of the confined fields on pair-production rates are exhibited. A simple variable change in all formulas leads to results for electric fields depending on time rather than space. In addition, we discuss tunneling processes in which empty atomic bound states are spontaneously filled by negative-energy electrons from the vacuum under positron emission. In particular, we calculate the rate at which the atomic levels of a bare nucleus of finite size $r_{\rm n}$ and large $Z\gg 1$ are filled by spontaneous pair creation.Comment: 33 pages and 9 figures. to appear in Phys. Rev.

Modeling software design diversity

Design diversity has been used for many years now as a means of achieving a degree of fault tolerance in software-based systems. Whilst there is clear evidence that the approach can be expected to deliver some increase in reliability compared with a single version, there is not agreement about the extent of this. More importantly, it remains difficult to evaluate exactly how reliable a particular diverse fault-tolerant system is. This difficulty arises because assumptions of independence of failures between different versions have been shown not to be tenable: assessment of the actual level of dependence present is therefore needed, and this is hard. In this tutorial we survey the modelling issues here, with an emphasis upon the impact these have upon the problem of assessing the reliability of fault tolerant systems. The intended audience is one of designers, assessors and project managers with only a basic knowledge of probabilities, as well as reliability experts without detailed knowledge of software, who seek an introduction to the probabilistic issues in decisions about design diversity

Angular distributions of scattered excited muonic hydrogen atoms

Differential cross sections of the Coulomb deexcitation in the collisions of excited muonic hydrogen with the hydrogen atom have been studied for the first time. In the framework of the fully quantum-mechanical close-coupling approach both the differential cross sections for the $nl \to n'l'$ transitions and $l$-averaged differential cross sections have been calculated for exotic atom in the initial states with the principle quantum number $n=2 - 6$ at relative motion energies $E_{\rm {cm}}=0.01 - 15$ eV and at scattering angles $\theta_{\rm {cm}}=0 - 180^{\circ}$. The vacuum polarization shifts of the $ns$-states are taken into account. The calculated in the same approach differential cross sections of the elastic and Stark scattering are also presented. The main features of the calculated differential cross sections are discussed and a strong anisotropy of cross sections for the Coulomb deexcitation is predicted.Comment: 5 pages, 9 figure