arXiv subjects

Fritz Haake

Publications and source records attributed to Fritz Haake.

39 records · Page 3Linked to original sources

arXiv1997

Semiclassical spectra from periodic-orbit clusters in a mixed phase space

We determine semiclassical quasienergy spectra from periodic orbits for a system with a mixed phase space, the kicked top. Throughout the transition from integrability to well developed chaos the standard error incurred for the quasienergies is a small percentage of their mean spacing. Such fine accuracy does not even require the angular momentum quantum number $j\propto 1/\hbar$ to be large; it already prevails for $j=1,2,3$. To open the way towards a reliable spectrum the conventional trace formula à la Gutzwiller has to be extended by including (i) ghosts, i.e., complex predecessors of bifurcating orbits and the possibility of Stokes transitions undergone by such complex orbits and (ii) collective contributions of clusters of periodic orbits near bifurcations. Even bifurcations of codimension higher than one must be reckoned with by accounting for the clusters involved through the appropriate diffraction integrals.

chao-dyn↗

arXiv1996

Secular determinants of random unitary matrices

We consider the characteristic polynomials of random unitary matrices $U$ drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The variances of these ``secular coefficients'' are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent $β$. The latter secular coefficients are related to the traces of powers of $U$ by Newton's well-known formulae. While the traces tend to have Gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large, the secular coefficients exhibit strong mutual correlations due to Newton's mixing of traces to coefficients. These results might become relevant for current efforts at combining semiclassics and random-matrix theory in quantum treatments of classically chaotic dynamics.

chao-dyn↗

arXiv1994

Cat States and Single Runs for the Damped Harmonic Oscillator

We discuss the fate of initial states of the cat type for the damped harmonic oscillator, mostly employing a linear version of the stochastic Schrödinger equation. We also comment on how such cat states might be prepared and on the relation of single realizations of the noise to single runs of experiments.

gr-qc↗