Energy evolution in time-dependent harmonic oscillator with arbitrary external forcing

The classical Hamiltonian system of time-dependent harmonic oscillator driven by the arbitrary external time-dependent force is considered. Exact analytical solution of the corresponding equations of motion is constructed in the framework of the technique (Robnik M, Romanovski V G, J. Phys. A: Math. Gen. {\bf 33} (2000) 5093) based on WKB approach. Energy evolution for the ensemble of uniformly distributed w.r.t. the canonical angle initial conditions on the initial invariant torus is studied. Exact expressions for the energy moments of arbitrary order taken at arbitrary time moment are analytically derived. Corresponding characteristic function is analytically constructed in the form of infinite series and numerically evaluated for certain values of the system parameters. Energy distribution function is numerically obtained in some particular cases. In the limit of small initial ensemble's energy the relevant formula for the energy distribution function is analytically derived.Comment: 16 pages, 5 figure

WKB expansion for the angular momentum and the Kepler problem: from the torus quantization to the exact one

We calculate the WKB series for the angular momentum and the non--relativistic 3-dim Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.SqComment: 16 pages plain Latex file, no figures. submitted to J. Phys.

Some generic properties of level spacing distributions of 2D real random matrices

We study the level spacing distribution $P(S)$ of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for $P(S)$ is derived and its behaviour close to S=0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For the {\em symmetric} real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent $\rho$ is always $\rho = 1$, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of $S$, the level spacing distribution $P(S)$ is a fractional exponent pawer law at small $S$. The tail of $P(S)$ depends on further details of the matrix element statistics. We explicitly work out four cases: the constant (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for $P(S)$ near zero value times an exponential tail.Comment: 21 pages, no figures, submitted to Zeitschrift fuer Naturforschung