Internal and External Resonances of Dielectric Disks

Circular microresonators (microdisks) are micron sized dielectric disks embedded in a material of lower refractive index. They possess modes with complex eigenvalues (resonances) which are solutions of analytically given transcendental equations. The behavior of such eigenvalues in the small opening limit, i.e. when the refractive index of the cavity goes to infinity, is analysed. This analysis allows one to clearly distinguish between internal (Feshbach) and external (shape) resonant modes for both TM and TE polarizations. This is especially important for TE polarization for which internal and external resonances can be found in the same region of the complex wavenumber plane. It is also shown that for both polarizations, the internal as well as external resonances can be classified by well defined azimuthal and radial modal indices.Comment: 5 pages, 8 figures, pdflate

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

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

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

Semiclassical expansion of parametric correlation functions of the quantum time delay

We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the two-point correlation function, the form factor $K(\tau,x,y,M)$, that depends on the number of open channels $M$, a non-symmetry breaking parameter $x$, and a symmetry breaking parameter $y$. Several terms in the Taylor expansion about $\tau=0$, which depend on all parameters, are shown to be identical to those obtained from Random Matrix Theory.Comment: 21 pages, no figure

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.

On the Accuracy of the Semiclassical Trace Formula

The semiclassical trace formula provides the basic construction from which one derives the semiclassical approximation for the spectrum of quantum systems which are chaotic in the classical limit. When the dimensionality of the system increases, the mean level spacing decreases as $\hbar^d$, while the semiclassical approximation is commonly believed to provide an accuracy of order $\hbar^2$, independently of d. If this were true, the semiclassical trace formula would be limited to systems in d <= 2 only. In the present work we set about to define proper measures of the semiclassical spectral accuracy, and to propose theoretical and numerical evidence to the effect that the semiclassical accuracy, measured in units of the mean level spacing, depends only weakly (if at all) on the dimensionality. Detailed and thorough numerical tests were performed for the Sinai billiard in 2 and 3 dimensions, substantiating the theoretical arguments.Comment: LaTeX, 31 pages, 14 figures, final version (minor changes

Mittelfristige Ergebnisse der Vastus-medialis-obliquus-Plastik bei lateraler Patellaluxation

Zusammenfassung: In Langzeitergebnissen nach traditionellen Operationsverfahren distalen Realignements für Patellaluxationen wie z.B. der Tuberositasosteotomie wird eine hohe Rate an Femoropatellararthrosen gefunden, sodass ein operatives Vorgehen noch heute kontrovers diskutiert wird. In der Literatur scheinen die Verfahren mit dynamischem proximalem Realignement eine geringere Arthroserate, aber bisweilen höhere Reluxationsrate aufzuweisen. Unlängst wurde der M.vastus medialis obliquus (VMO) in anatomischen und biomechanischen Studien als eine der entscheidenden proximalen stabilisierenden Strukturen bei lateralen Patellaluxationen identifiziert. Zwischen 1994 und 2003 wurden 28Patienten (Durchschnittsalter 21,5Jahre) mit einer VMO-Plastik bei lateraler Patellaluxation operativ versorgt. Die Technik wurde für die meisten Ätiologien einer femoropatellären Instabilität angewandt. Bei dieser proximalen Weichteilkorrektur wird die sehnige Einstrahlung des VMO von der Patella abgelöst. Anschließend wird die Sehne 10-15mm distalisierend an der Patella über MITEK-Anker reinseriert. Postoperativ ist eine Vollbelastung in Streckung möglich. Ein aktives Auftrainieren des Vastus medialis beginnt 6Wochen nach der Operation. 27Patienten wurden klinisch und radiologisch im Jahre 2004 nachkontrolliert, durchschnittlich 5Jahre nach der Operation. 83% gaben ein exzellentes oder gutes Resultat an, 10% waren zufrieden und 7% unzufrieden. Der durchschnittliche Lysholm-Knie-Score betrug 83,1Punkte. Zwei Patienten erlitten eine Reluxation (7%). Die postoperativen Röntgenbilder zeigten eine signifikante Verbesserung des Kongruenzwinkels auch noch nach vielen Jahren. In 89% der Fälle wurde keine oder eine nur geringe Femoropatellararthrose beobachtet. Die präsentierten Fünfjahresergebnisse sind bezüglich Patientenzufriedenheit mit anderen Verfahren proximalen und distalen Realignements vergleichbar. Die Reluxationsrate ist unterdurchschnittlich. Die bisherige niedrige Rate an Femoropatellararthrose nach durchschnittlich 5Jahren erscheint im Vergleich mit den Arthroseraten des rigiden, distalen Realignements hinsichtlich zukünftiger Langzeitergebnisse vielversprechend und wird auf den minimalen Eingriff in das physiologische Gelenkspiel und auf die Wiederherstellung der verletzten Anatomie zurückgeführt. Die Idee der proximalen dynamischen Stabilisierung und das Angreifen am Ursprung der Pathologie wird in den Erkenntnissen aktueller anatomischer und biomechanischer Untersuchungen bestätigt, was diese relativ guten Ergebnisse erklären mag. Über- und Unterkorrekturen der Weichteile können zurzeit kompensiert werden. Die oben genannten traditionellen und rigideren Operationsmethoden erlauben eine solche Kompensation nicht in diesem Ausmaß und können so zu präarthrotischer Überbelastung des medialen Femoropatellar- und Femorotibialgelenks führe

Quantal Consequences of Perturbations Which Destroy Structurally Unstable Orbits in Chaotic Billiards

Non-generic contributions to the quantal level-density from parallel segments in billiards are investigated. These contributions are due to the existence of marginally stable families of periodic orbits, which are structurally unstable, in the sense that small perturbations, such as a slight tilt of one of the segments, destroy them completely. We investigate the effects of such perturbation on the corresponding quantum spectra, and demonstrate them for the stadium billiard

Semiclassical approach to discrete symmetries in quantum chaos

We use semiclassical methods to evaluate the spectral two-point correlation function of quantum chaotic systems with discrete geometrical symmetries. The energy spectra of these systems can be divided into subspectra that are associated to irreducible representations of the corresponding symmetry group. We show that for (spinless) time reversal invariant systems the statistics inside these subspectra depend on the type of irreducible representation. For real representations the spectral statistics agree with those of the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT), whereas complex representations correspond to the Gaussian Unitary Ensemble (GUE). For systems without time reversal invariance all subspectra show GUE statistics. There are no correlations between non-degenerate subspectra. Our techniques generalize recent developments in the semiclassical approach to quantum chaos allowing one to obtain full agreement with the two-point correlation function predicted by RMT, including oscillatory contributions.Comment: 26 pages, 8 Figure