arXiv subjects

Gregor Tanner

Publications and source records attributed to Gregor Tanner.

40 records · Page 3Linked to original sources

arXiv2000

Periodic orbit action correlations in the Baker map

Periodic orbit action correlations are studied for the piecewise linear, area-preserving Baker map. Semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum Baker map suggest the existence of universal periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius type operator. This makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map. Periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error. Exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation.

nlin.CD↗

arXiv2000

Spectral statistics for unitary transfer matrices of binary graphs

Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalise this approach by considering arbitrary, directed graphs with unitary transfer matrices. An exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes. A special class of graphs, so-called binary graphs, is studied in more detail. For these, the conditions for periodic orbit pairs to be correlated (including correlations due to the unitarity of the transfer matrix) can be given explicitly. Using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low--dimensional cases. Gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs.

nlin.CD↗

arXiv1996

How Chaotic is the Stadium Billiard? A Semiclassical Analysis

The impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described, semiclassically, by the Gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits. I show that this belief is erroneous. The Gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits. Unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the Green function. Semiclassical contributions to the trace show an $\hbar$ - dependent transition from hard chaos to integrable behavior for trajectories approaching the bouncing ball orbits. A whole region in phase space surrounding the marginal stable family acts, semiclassically, like a stable island with boundaries being explicitly $\hbar$-dependent. The localized bouncing ball states found in the billiard derive from this semiclassically stable island. The bouncing ball orbits themselves, however, do not contribute to individual eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing ball eigenstates in the stadium can be derived. The stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics.

chao-dyn↗

arXiv1996

The semiclassical resonance spectrum of Hydrogen in a constant magnetic field

We present the first purely semiclassical calculation of the resonance spectrum in the Diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with $L_z =0$. The classical system is unbound and completely chaotic for a scaled energy $ε\sim E B^{-2/3}$ larger than a critical value $ε_c >0$. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron--nucleus distance causes divergences in the periodic orbit formula. Using a regularisation technique introduced in [G.\ Tanner and D.\ Wintgen, Phys.~Rev.~Lett.~{\bf 75}, 2928 (1995)] together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi Einstein--Brillouin--Keller (QEBK) quantisation is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg--like series converging towards the different Landau thresholds.

chao-dyn↗