Semiclassical Transition from an Elliptical to an Oval Billiard

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when $\hbar$ is small in comparison to relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In the present paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approximations for bifurcations of periodic orbits. Relations between different uniform approximations are discussed.Comment: LaTeX, 33 pages including 10 PostScript figures, submitted to J. Phys.

Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems

We consider the semiclassical approximation to the spectral form factor K(tau) for two-dimensional uniformly hyperbolic systems, and derive the first off-diagonal correction for small tau. The result agrees with the tau^2-term of the form factor for the GOE random matrix ensemble.Comment: 8 pages, 3 figure

The semiclassical relation between open trajectories and periodic orbits for the Wigner time delay

The Wigner time delay of a classically chaotic quantum system can be expressed semiclassically either in terms of pairs of scattering trajectories that enter and leave the system or in terms of the periodic orbits trapped inside the system. We show how these two pictures are related on the semiclassical level. We start from the semiclassical formula with the scattering trajectories and derive from it all terms in the periodic orbit formula for the time delay. The main ingredient in this calculation is a new type of correlation between scattering trajectories which is due to trajectories that approach the trapped periodic orbits closely. The equivalence between the two pictures is also demonstrated by considering correlation functions of the time delay. A corresponding calculation for the conductance gives no periodic orbit contributions in leading order.Comment: 21 pages, 5 figure

Trace formula for a dielectric microdisk with a point scatterer

Two-dimensional dielectric microcavities are of widespread use in microoptics applications. Recently, a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the cavity. In the present paper we extend this trace formula to a dielectric disk with a small scatterer. This system has been introduced for microlaser applications, because it has long-lived resonances with strongly directional far field. We show that its resonance spectrum contains signatures not only of periodic rays, but also of diffractive rays that occur in Keller's geometrical theory of diffraction. We compare our results with those for a closed cavity with Dirichlet boundary conditions.Comment: 39 pages, 18 figures, pdflate

Semiclassical Theory of Chaotic Quantum Transport

We present a refined semiclassical approach to the Landauer conductance and Kubo conductivity of clean chaotic mesoscopic systems. We demonstrate for systems with uniformly hyperbolic dynamics that including off-diagonal contributions to double sums over classical paths gives a weak-localization correction in quantitative agreement with results from random matrix theory. We further discuss the magnetic field dependence. This semiclassical treatment accounts for current conservation.Comment: 4 pages, 1 figur

Single chain dimers of MASH-1 bind DNA with enhanced affinity

By designing recombinant genes containing tandem copies of the coding region of the BHLH domain of MASH-1 (MASH-BHLH) with intervening DNA sequences encoding linker sequences of 8 or 17 amino acids, the two subunits of the MASH dimer have been connected to form the single chain dimers MM8 and MM17. Despite the long and flexible linkers which connect the C-terminus of the first BHLH subunit to the N-terminus of the second, a distance of ∼55 Å, the single chain dimers could be produced in Escherichia coli at high levels. MM8 and MM17 were monomeric and no ‘cross-folding' of the subunits was observed. CD spectroscopy revealed that, like wild-type MASH-BHLH, MM8 and MM17 adopt only partly folded structures in the absence of DNA, but undergo a folding transition to a mainly α-helical conformation on DNA binding. Titrations by electrophoretic mobility shift assays revealed that the affinity of the single chain dimers for E box-containing DNA sequences was increased ∼10-fold when compared with wild-type MASH-BHLH. On the other hand, the affinity for heterologous DNA sequences was increased only 5-fold. Therefore, the introduction of the peptide linker led to a 4-fold increase in DNA binding specificity from −0.14 to −0.57 kcal/mo

Particle creation and annihilation at interior boundaries:One-dimensional models

We describe creation and annihilation of particles at external sources in one spatial dimension in terms of interior-boundary conditions (IBCs). We derive explicit solutions for spectra, (generalised) eigenfunctions, as well as Green functions, spectral determinants, and integrated spectral densities. Moreover, we introduce a quantum graph version of IBC-Hamiltonians.Comment: 32 page

Semiclassical universality of parametric spectral correlations

We consider quantum systems with a chaotic classical limit that depend on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor $K(\tau,x)$ which depends on a scaled parameter difference $x$. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small $\tau$ expansion of the form factor agrees with Random Matrix Theory for systems with and without time reversal symmetry.Comment: 18 pages, no figure

Spectral statistics in chaotic systems with a point interaction

We consider quantum systems with a chaotic classical limit that are perturbed by a point-like scatterer. The spectral form factor K(tau) for these systems is evaluated semiclassically in terms of periodic and diffractive orbits. It is shown for order tau^2 and tau^3 that off-diagonal contributions to the form factor which involve diffractive orbits cancel exactly the diagonal contributions from diffractive orbits, implying that the perturbation by the scatterer does not change the spectral statistic. We further show that parametric spectral statistics for these systems are universal for small changes of the strength of the scatterer.Comment: LaTeX, 21 pages, 7 figures, small corrections, new references adde