Inclusion of Diffraction Effects in the Gutzwiller Trace Formula

The Gutzwiller trace formula is extended to include diffraction effects. The new trace formula involves periodic rays which have non-geometrical segments as a result of diffraction on the surfaces and edges of the scatter.Comment: 4 pages, LaTeX, 1 ps figur

The Measurement of Sexual Selection Using Bateman's Principles: An Experimental Test in the Sex-Role-Reversed Pipefish Syngnathus typhle.

Angus J. Bateman's classic study of sexual selection in Drosophila melanogaster has had a major influence on the development of sexual selection theory. In some ways, Bateman's study has served a catalytic role by stimulating debate on sex roles, sexual conflict and other topics in sexual selection. However, there is still considerable disagreement regarding whether or not "Bateman's principles" are helpful in the study of sexual selection. Here, we test the idea that Bateman's principles provide the basis for a useful method to quantify and compare mating systems. In this study, we focus on the sex-role-reversed pipefish Syngnathus typhle as a model system to study the measurement of sexual selection. We set up artificial breeding assemblages of pipefish in the laboratory and used microsatellite markers to resolve parentage. Three different sex-ratio treatments (female-biased, even and male-biased) were used to manipulate the expected intensity of sexual selection. Measures of the mating system based on Bateman's principles were calculated and compared to the expected changes in the intensity of sexual selection. We also compare the results of this study to the results of a similar study of Bateman's principles in the rough-skinned newt, a species with conventional sex roles. The results of this experiment show that measures of the mating system based on Bateman's principles do accurately capture the relative intensities of sexual selection in the different treatments and species. Thus, widespread use of Bateman's principles to quantify mating systems in nature would facilitate comparative studies of sexual selection and mating system evolution

The Bateman gradient and the cause of sexual selection in a sexrolereversed pipefish

As a conspicuous evolutionary mechanism, sexual selection has received much attention from theorists and empiricists. Although the importance of the mating system to sexual selection has long been appreciated, the precise relationship remains obscure. In a classic experimental study based on parentage assessment using visible genetic markers, more than 50 years ago A. J. Bateman proposed that the cause of sexual selection in Drosophila is 'the stronger correlation, in males (relative to females), between number of mates and fertility (number of progeny)'. Half a century later, molecular genetic techniques for assigning parentage now permit mirror-image experimental tests of the 'Bateman gradient' using sex-role-reversed species. Here we show that, in the male-pregnant pipefish Syngnathus typhle, females exhibit a stronger positive association between number of mates and fertility than do males and that this relationship responds in the predicted fashion to changes in the adult sex ratio. These findings give empirical support to the idea that the relationship between mating success and number of progeny, as characterized by the Bateman gradient, is a central feature of the genetic mating system affecting the strength and direction of sexual selection

Small Disks and Semiclassical Resonances

We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the order of the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the order of the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks.Comment: Final published version - some changes in the discussion and the labels on one figure are correcte

Geometrical theory of diffraction and spectral statistics

We investigate the influence of diffraction on the statistics of energy levels in quantum systems with a chaotic classical limit. By applying the geometrical theory of diffraction we show that diffraction on singularities of the potential can lead to modifications in semiclassical approximations for spectral statistics that persist in the semiclassical limit $\hbar \to 0$. This result is obtained by deriving a classical sum rule for trajectories that connect two points in coordinate space.Comment: 14 pages, no figure, to appear in J. Phys.

Uniform approximation for diffractive contributions to the trace formula in billiard systems

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.Comment: LaTeX, 41 pages including 12 PostScript figures, submitted to Phys. Rev.

Three disks in a row: A two-dimensional scattering analog of the double-well problem

We investigate the scattering off three nonoverlapping disks equidistantly spaced along a line in the two-dimensional plane with the radii of the outer disks equal and the radius of the inner disk varied. This system is a two-dimensional scattering analog to the double-well-potential (bound state) problem in one dimension. In both systems the symmetry splittings between symmetric and antisymmetric states or resonances, respectively, have to be traced back to tunneling effects, as semiclassically the geometrical periodic orbits have no contact with the vertical symmetry axis. We construct the leading semiclassical ``creeping'' orbits that are responsible for the symmetry splitting of the resonances in this system. The collinear three-disk-system is not only one of the simplest but also one of the most effective systems for detecting creeping phenomena. While in symmetrically placed n-disk systems creeping corrections affect the subleading resonances, they here alone determine the symmetry splitting of the 3-disk resonances in the semiclassical calculation. It should therefore be considered as a paradigm for the study of creeping effects. PACS numbers: 03.65.Sq, 03.20.+i, 05.45.+bComment: replaced with published version (minor misprints corrected and references updated); 23 pages, LaTeX plus 8 Postscript figures, uses epsfig.sty, espf.sty, and epsf.te