An analytical approach to the multiply scattered light in the optical images of the extensive air showers of ultra-high energies

One of the methods for studying the highest energy cosmic rays is to measure the fluorescence light emitted by the extensive air showers induced by them. To reconstruct a shower cascade curve from measurements of the number of photons arriving from the subsequent shower track elements it is necessary to take into account the multiple scatterings that photons undergo on their way from the shower to the detector. In contrast to the earlier Monte-Carlo work, we present here an analytical method to treat the Rayleigh and Mie scatterings in the atmosphere. The method consists in considering separately the consecutive 'generations' of the scattered light. Starting with a point light source in a uniform medium, we then examine a source in a real atmosphere and finally - a moving source (shower) in it. We calculate the angular distributions of the scattered light superimposed on the not scattered light registered from a shower at a given time. The analytical solutions (although approximate) show how the exact numerical results should be parametrised what we do for the first two generations (the contribution of the higher ones being small). Not allowing for the considered effect may lead to an overestimation of shower primary energy by ~15% and to an underestimation of the primary particle mass.Comment: 23 pages, 18 figures, submited to Astroparticle Physic

Rectangular Well as Perturbation

We discuss a finite rectangular well as a perturbation for the infinite one with a depth $\lambda^2$ of the former as a perturbation parameter. In particular we consider a behaviour of energy levels in the well as functions of complex $\lambda$. It is found that all the levels of the same parity are defined on infinitely sheeted Riemann surfaces which topological structures are described in details. These structures differ considerably from those found in models investigated earlier. It is shown that perturbation series for all the levels converge what is in contrast with the known results of Bender and Wu. The last property is shown to hold also for the finite rectangular well with Dirac delta barier as a perturbation considered earlier by Ushveridze.Comment: 19 pages, 5 Postscript figures, uses psfig.st

Superscar states in rational polygon billiards - a reality or an illusion?

The superscars phenomena (Heller, E.J., Phys. Rev. Lett. 53, (1984) 1515) in the rational polygon billiards (RPB) are analysed using the high energy semiclassical wave functions (SWF) built on classical trajectories forming skeletons. Considering examples of the pseudointegrable billiards such as the Bogomolny-Schmit triangle, the parallelogram and the L-shape billiards as well as the integrable rectangular one the constructed SWFs allow us to verify the idea of Bogomolny and Schmit (Phys. Rev. Lett. 92 (2004) 244102) of SWFs (superscars) propagating along periodic orbit channels (POC) and vanishing outside of them. It is shown that the superscars effects in RPB appear as natural properties of SWFs built on the periodic skeletons. The latter skeletons are commonly present in RPB and are always composed of POCs. The SWFs built on the periodic skeletons satisfy all the basic principles of the quantum mechanics contrary to the superscar states of Bogomolny and Schmit which break them. Therefore the superscars effects need not to invoke the idea of the superscar states of Bogomolny and Schmit at least in the cases considered in our paper.Comment: 33 pages, 5 figure

Fundamental solution method applied to time evolution of two energy level systems: exact and adiabatic limit results

A method of fundamental solutions has been used to investigate transitions in two energy level systems with no level crossing in a real time. Compact formulas for transition probabilities have been found in their exact form as well as in their adiabatic limit. No interference effects resulting from many level complex crossings as announced by Joye, Mileti and Pfister (Phys. Rev. {\bf A44} 4280 (1991)) have been detected in either case. It is argued that these results of this work are incorrect. However, some effects of Berry's phases are confirmed.Comment: LaTeX2e, 23 pages, 8 EPS figures. Style correcte

Semiclassical wave functions and energy spectra in polygon billiards

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. Unfolding rational polygon billiards (RPB) into corresponding Riemann surfaces (RS) periodic structures of the latter are demonstrated with 2g independent periods on the respective multitori with g as their genuses. However it is the two dimensional real space of the real linear combinations of these periods which is used for quantizing RPB. A class of doubly rational polygon billiards (DRPB) is distinguished for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is shown that semiclassical quantization of both the classical momenta and the energy spectra are determined completely by periodic structure of the corresponding RS. Each RS is then reduced to elementary polygon patterns (EPP) as its basic periodic elements. Each such EPP can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then constructed on EPP. The SWF for DRPB appear to be exact. They satisfy the Dirichlet, the Neumannn or the mixed boundary conditions. Not every mixing is allowed however and a respective incompleteness of SWF is discussed. Dens families of DRPB are used for approximate semiclassical quantization of RPB. General rational polygons are quantized by approximating them by DRPB. An extension of the formalism to irrational polygons is described as well. The semiclassical approximations constructed in the paper are controlled by general criteria of the eigenvalue theory. A relation between the superscar solutions and SWF constructed in the paper is also discussed.Comment: 34 pages, 5 figure