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We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared to the wave length 2 pi/k. Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact KKR determinant, which we generalize here for the A_1 subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer c=J_0(kR)/Y_0(kR). Surprisingly, this can take any value between plus and minus infinity within the range of validity of the diffractive approximation kR <<4. We study the statistics exhibited by spectra for fixed values of c. It is Poissonian for |c|=infinity, provided the disk is placed inside a rectangle whose sides obeys some constraints. For c=0 we find a good agreement of the level spacing distribution with GOE, whereas the form factor and two-point correlation function are similar but exhibit larger deviations. By varying the parameter c from 0 to infinity the level statistics interpolates smoothly between these limiting cases., Comment: 17 pages LaTeX, 5 postscript figures, submitted to J. Phys. A: Math. Gen

The topological pressure introduced by Ruelle and similar quantities describe dynamical multifractal properties of dynamical systems. These are important characteristics of mesoscopic systems in the classical regime. Original definition of these quantities are based on the symbolic description of the dynamics. It is hard or impossible to find symbolic description and generating partition to a general dynamical system, therefore these quantities are often not accessible for further studies. Here we present a new method by which the symbolic description can be omitted. We apply the method for a mixing and an intermittent system., Comment: 8 pages LaTeX with revtex.sty, the 4 postscript figures are included using psfig.tex to appear in PRL

The diffraction trace formula ({\em Phys. Rev. Lett.} {\bf 73}, 2304 (1994)) and spectral determinant are tested on the open three disk scattering system. The system contains a generic and exponentially growing number of diffraction periodic orbits. In spite of this it is shown that even the scattering resonances with large imaginary part can be reproduced semiclassicaly. The non-trivial interplay of the diffraction periodic orbits with the usual geometrical orbits produces the fine structure of the complicated spectrum of scattering resonances, which are beyond the resolution of the conventional periodic orbit theory., Comment: Latex article + 3 ps figures

We show, that the canonical invariant part of $\hbar$ corrections to the Gutzwiller trace formula and the Gutzwiller-Voros spectral determinant can be computed by the Bohr-Sommerfeld quantization rules, which usually apply for integrable systems. We argue that the information content of the classical action and stability can be used more effectively than in the usual treatment. We demonstrate the improvement of precision on the example of the three disk scattering system., Comment: revtex

A quantum generalization of the semiclassical theory of Gutzwiller is given. The new formulation leads to systematic orbit-by-orbit inclusion of higher $\hbar$ contributions to the spectral determinant. We apply the theory to billiard systems, and compare the periodic orbit quantization including the first $\hbar$ contribution to the exact quantum mechanical results., Comment: revtex

An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and which correspond to creping waves. A new family of resonances in the two disk scattering system can be well described which is completely missing if only the traditional periodic orbits are used., Comment: 4 pages, latex/REVTEX, 2 PS figures

We show that the daily average air humidity fluctuations exhibit non-trivial $1/f^{\alpha}$ behaviour which different from the spectral properties of other meteorological quantities. This feature and the fractal spatial strucure found in clouds make it plausible to regard air humidity fluctuations as a manifestation of self-organized criticality. We give arguments why the dynamics in air humidity can be similar to those in sandpile models of SOC., Comment: 3 pages, latex rewtex, 6 short Post Script files attached