Metals with excitonic instabilities are multiband systems with significant electron-electron interaction. The electronic transport in such systems is affected by collective fluctuations of the electrons, leading to anomalous features in the measured transport coefficients. Many of these anomalies have not been well understood because the transport mechanisms in these systems tend to be rather complex. The complexity arises, on the one hand, from the multiband nature and, on the other, from the anisotropic scattering of electrons accompanied by emitting or absorbing collective fluctuations. Previous works considering scattering due to collective fluctuations have mainly focused on single-band systems, for example in the context of the normal-state transport in cuprates. The recent discovery of high-temperature superconductivity in iron pnictides has renewed the interest in multiband systems. Exploring the transport mechanisms in multiband systems, I find some interesting new aspects, which do not occur in single-band systems. In particular, anisotropic scattering in a model with electronlike and holelike Fermi surfaces can lead to a negative conductivity contribution of the minority carriers, i.e., in an electric field, the minority carriers drift in the direction opposite of what one would expect based on their charge. I show that this effect can explain a reduced magnetoresistance in connection with an enhanced Hall coefficient, which has been measured in pnictides. Of particular interest are multiband models with hot spots on the Fermi surface, in part because of their relevance for the iron pnictides. Hot spots are states with enhanced scattering and therefore reduced excitation lifetimes. In single-band systems, the hot spots are found to have a much lower contribution to the total conductivity than other parts of the Fermi surface, which leads to the so-called hot-spot structure. I show that in the multiband case, the conductivity contributions are much more isotropic around the Fermi surface so that hot spots contribute to transport with a similar strength as other parts of the Fermi surface. I discuss this effect on the basis of an approximate analytical solution of the transport problem and numerically calculate the temperature dependence of several transport coefficients. It turns out that in the nematic phase of iron pnictides, the unexpectedly strong conductivity contribution of hot spots can explain the puzzling behavior of the resistive anisotropy. I show that the experimental observations can be explained within a scenario in which the anisotropy is mainly due to the broken symmetry of the spin-fluctuation spectrum in the nematic phase. In the spin-density-wave state, strongly anisotropic scattering can arise due to the propagating magnons. Using a two-band model relevant for iron pnictides, I find that this scattering can lead to an unusual interruption of the orbital motion of electrons in the magnetic field. As a consequence, the low-field magnetoresistance is linear with an alternating sign of the slope as a function of the direction of the current. In strong magnetic fields, the interrupted orbital motion makes the system unstable, which is characterized by a drop of the resistivity to zero.

We consider scattering in $R^n$, $nge 2$, described by the Schr"odinger operator $P(h)=-h^2Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $hto 0$ of the scattering amplitude $f(omega_-,omega_+;lambda,h)$ $omega_+neqomega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity. / Wir betrachten Streuung in $R^n$, $nge 2$, beschrieben durch den Schr"odinger operator $P(h)=-h^2Delta+V$, wo $V$ ein kurzreichweitiges Potential ist. Mit Hilfe von Maslov Theorie erhalten wir eine geometrische Formel fuer die semiklassische Asymptotik ($hto 0$) der Streuamplitude $f(omega_-,omega_+;lambda,h)$ ($omega_+neqomega_-$) welche auch bei Vorhandensein von Fokalpunkten bei Unendlich (Kaustiken) gueltig bleibt.

Semiclassical initial value representations (SC-IVRs) are popular methods for an approximate description of the quantum dynamics of atomic and molecular systems. A very efficient special case is the propagator by Herman and Kluk, which will be the basis for the investigations in this work. It consists of a phase space integration over initial conditions of classical trajectories which are guiding Gaussian wavepackets. A complex phase factor in the integrand allows for interference between different trajectories, which leads to soft quantum effects being naturally included in the description. The underlying classical trajectories allow for an approximate description of the dynamics of large quantum systems that are inaccessible for a full quantum propagation. Moreover, they also provide an intuitive understanding of quantum phenomena in terms of classical dynamics. The main focus of this work is on further approximations to Herman-Kluk propagation whose applicability to complex dynamics is limited by the number of trajectories that are needed for numerical convergence of the phase space integration. The central idea for these approximations is the semiclassical hybrid formalism which utilizes the costly Herman-Kluk propagator only for a small number of system degrees of freedom (DOFs). The remaining environmental DOFs are treated on the level of Heller's thawed Gaussian wavepacket dynamics, a single trajectory method which is exact only for at most harmonic potentials. If the environmental DOFs are weakly coupled and therefore close to their potential minimum, this level of accuracy is sufficient to account for their effect on the system. Thus, the hybrid approximation efficiently combines accuracy and low numerical cost. As a central theoretical result, we apply this hybrid idea to a time-averaging scheme to arrive at a method for the calculation of vibrational spectra of molecules that is both accurate and efficient. This time-averaged hybrid propagation is then used to study the vibrational dynamics of an iodine-like Morse oscillator bilinearly coupled to a Caldeira-Leggett bath of harmonic oscillators. We first validate the method by comparing it to full quantum and Herman-Kluk propagation for appropriately sized environments. After having established its accuracy, we include more bath DOFs to investigate the influence of the Caldeira-Leggett counter term on the shift of the vibrational levels of the Morse oscillator. As a result, we find out that a redshift, which is observed experimentally for, e.g., iodine in a rare gas matrix, occurs only if the counter term is not included in the Hamiltonian. We then move away from the model bath and on to a realistic, experimentally relevant environment consisting of krypton atoms. We put the iodine molecule into a cluster of 17 krypton atoms and investigate the loss of coherence of the iodine vibration upon coupling to just a few normal coordinates of the bath. These modes with the same symmetry as the iodine vibration turn out to be sufficient to reproduce the expected qualitative dependence on bath temperature and initial state of the iodine molecule. With these few normal modes, a full quantum calculation yields values for coherence loss rates that are close to experimental results. Furthermore, a comparison to semiclassical calculations with more bath modes included confirms the importance of the few highly symmetric normal coordinates. Then, we apply the time-averaged hybrid formalism once more to calculate the vibrational spectrum of the iodine molecule in this now anharmonic krypton environment. Using a krypton matrix instead of a cluster geometry, we find the correct qualitative and also quite good quantitative agreement for the shift of the iodine potential. Finally, we will investigate a more fundamental question, namely, if SC-IVRs contain the spin effects due to the Pauli exclusion principle. To this end, we apply a number of SC-IVRs to the scattering of two electrons with initial states corresponding to either parallel or antiparallel spin. We compare the outcome to full quantum results and find that the difference is resolved by those methods that comprise multiple interfering trajectories.

The characteristics of optical modes in whispering-gallery cavities crucially depend on the underlying classical ray dynamics as they are subject to dynamical tunneling. In particular, classical nonlinear resonances lead to the hybridization of whispering-gallery modes spoiling their quality factors and decreasing their lifetimes via resonance-assisted tunneling. In this thesis we present an intuitive semiclassical description of resonance-assisted tunneling in deformed optical microdisks whose classical ray dynamics exhibits a mixed phase space. We find good agreement between semiclassically obtained decay rates of whispering-gallery modes and numerical solutions of the mode equation computed with the boundary element method. Moreover, we extend a perturbative description for weakly deformed microdisks with near-integrable ray dynamics to larger deformations and mixed phase spaces. This yields an accurate description of decay rates and of the near-field intensity distributions. Our approach is based on the approximation of the actual ray dynamics by an integrable Hamiltonian constructed in adiabatic action-angle coordinates. This allows for semiclassical quantization in order to determine the wave numbers of whispering-gallery modes as well as for a ray based description of their decay. The resonance-assisted coupling between individual modes is determined either perturbatively or semiclassically in terms of complex paths. / Flüstergaleriemoden in optischen Resonatoren zeigen dynamische Tunnelprozesse, welche maßgeblich von der zugrundeliegenden klassischen Strahlendynamik abhängen. Die Lebenszeit und die daraus resultierenden Gütefaktoren dieser Moden werden durch klassische nichtlineare Resonanzen und den Effekt des resonanzunterstützten Tunnelns verringert. Hierfür entwickeln wir eine intuitive semiklassische Beschreibung für den Fall deformierter optischer Kreiskavitäten, deren klassische Strahlendynamik einen gemischten Phasenraum aufweist. Die semiklassisch berechneten Zerfallsraten stimmen gut mit den numerischen Lösungen der Maxwell-Gleichungen, welche unter Nutzung der Randelementmethode ermittelt werden, überein. Darüber hinaus erweitern wir den Anwendungsbereich einer störungstheoretische Beschreibung von schwach deformierten Kavitäten hin zu größeren Deformationen. Dies ermöglicht nicht nur eine akkurate Vorhersage von Zerfallsraten, sondern auch die Beschreibung der Intensitätsverteilung von optischen Moden im Nahfeld. Unsere Methode basiert auf der Konstruktion von adiabatischen Winkel-Wirkungskoordinaten und der Approximation der Strahlendynamik durch ein integrables Hamiltonsches System. Mittels semiklassischer Quantisierung bestimmen wir damit die Wellenzahlen von Flüstergaleriemoden, deren Lebenszeit ferner durch ein strahlenbasiertes Modell beschrieben wird. Wir bestimmen die resonanzunterstützte Kopplung zwischen einzelnen solcher Moden sowohl mittels Störungstheorie als auch mittels klassischer komplexer Trajektorien.

Metals with excitonic instabilities are multiband systems with significant electron-electron interaction. The electronic transport in such systems is affected by collective fluctuations of the electrons, leading to anomalous features in the measured transport coefficients. Many of these anomalies have not been well understood because the transport mechanisms in these systems tend to be rather complex. The complexity arises, on the one hand, from the multiband nature and, on the other, from the anisotropic scattering of electrons accompanied by emitting or absorbing collective fluctuations. Previous works considering scattering due to collective fluctuations have mainly focused on single-band systems, for example in the context of the normal-state transport in cuprates. The recent discovery of high-temperature superconductivity in iron pnictides has renewed the interest in multiband systems. Exploring the transport mechanisms in multiband systems, I find some interesting new aspects, which do not occur in single-band systems. In particular, anisotropic scattering in a model with electronlike and holelike Fermi surfaces can lead to a negative conductivity contribution of the minority carriers, i.e., in an electric field, the minority carriers drift in the direction opposite of what one would expect based on their charge. I show that this effect can explain a reduced magnetoresistance in connection with an enhanced Hall coefficient, which has been measured in pnictides. Of particular interest are multiband models with hot spots on the Fermi surface, in part because of their relevance for the iron pnictides. Hot spots are states with enhanced scattering and therefore reduced excitation lifetimes. In single-band systems, the hot spots are found to have a much lower contribution to the total conductivity than other parts of the Fermi surface, which leads to the so-called hot-spot structure. I show that in the multiband case, the conductivity contributions are much more isotropic around the Fermi surface so that hot spots contribute to transport with a similar strength as other parts of the Fermi surface. I discuss this effect on the basis of an approximate analytical solution of the transport problem and numerically calculate the temperature dependence of several transport coefficients. It turns out that in the nematic phase of iron pnictides, the unexpectedly strong conductivity contribution of hot spots can explain the puzzling behavior of the resistive anisotropy. I show that the experimental observations can be explained within a scenario in which the anisotropy is mainly due to the broken symmetry of the spin-fluctuation spectrum in the nematic phase. In the spin-density-wave state, strongly anisotropic scattering can arise due to the propagating magnons. Using a two-band model relevant for iron pnictides, I find that this scattering can lead to an unusual interruption of the orbital motion of electrons in the magnetic field. As a consequence, the low-field magnetoresistance is linear with an alternating sign of the slope as a function of the direction of the current. In strong magnetic fields, the interrupted orbital motion makes the system unstable, which is characterized by a drop of the resistivity to zero.

Semiclassical initial value representations (SC-IVRs) are popular methods for an approximate description of the quantum dynamics of atomic and molecular systems. A very efficient special case is the propagator by Herman and Kluk, which will be the basis for the investigations in this work. It consists of a phase space integration over initial conditions of classical trajectories which are guiding Gaussian wavepackets. A complex phase factor in the integrand allows for interference between different trajectories, which leads to soft quantum effects being naturally included in the description. The underlying classical trajectories allow for an approximate description of the dynamics of large quantum systems that are inaccessible for a full quantum propagation. Moreover, they also provide an intuitive understanding of quantum phenomena in terms of classical dynamics. The main focus of this work is on further approximations to Herman-Kluk propagation whose applicability to complex dynamics is limited by the number of trajectories that are needed for numerical convergence of the phase space integration. The central idea for these approximations is the semiclassical hybrid formalism which utilizes the costly Herman-Kluk propagator only for a small number of system degrees of freedom (DOFs). The remaining environmental DOFs are treated on the level of Heller's thawed Gaussian wavepacket dynamics, a single trajectory method which is exact only for at most harmonic potentials. If the environmental DOFs are weakly coupled and therefore close to their potential minimum, this level of accuracy is sufficient to account for their effect on the system. Thus, the hybrid approximation efficiently combines accuracy and low numerical cost. As a central theoretical result, we apply this hybrid idea to a time-averaging scheme to arrive at a method for the calculation of vibrational spectra of molecules that is both accurate and efficient. This time-averaged hybrid propagation is then used to study the vibrational dynamics of an iodine-like Morse oscillator bilinearly coupled to a Caldeira-Leggett bath of harmonic oscillators. We first validate the method by comparing it to full quantum and Herman-Kluk propagation for appropriately sized environments. After having established its accuracy, we include more bath DOFs to investigate the influence of the Caldeira-Leggett counter term on the shift of the vibrational levels of the Morse oscillator. As a result, we find out that a redshift, which is observed experimentally for, e.g., iodine in a rare gas matrix, occurs only if the counter term is not included in the Hamiltonian. We then move away from the model bath and on to a realistic, experimentally relevant environment consisting of krypton atoms. We put the iodine molecule into a cluster of 17 krypton atoms and investigate the loss of coherence of the iodine vibration upon coupling to just a few normal coordinates of the bath. These modes with the same symmetry as the iodine vibration turn out to be sufficient to reproduce the expected qualitative dependence on bath temperature and initial state of the iodine molecule. With these few normal modes, a full quantum calculation yields values for coherence loss rates that are close to experimental results. Furthermore, a comparison to semiclassical calculations with more bath modes included confirms the importance of the few highly symmetric normal coordinates. Then, we apply the time-averaged hybrid formalism once more to calculate the vibrational spectrum of the iodine molecule in this now anharmonic krypton environment. Using a krypton matrix instead of a cluster geometry, we find the correct qualitative and also quite good quantitative agreement for the shift of the iodine potential. Finally, we will investigate a more fundamental question, namely, if SC-IVRs contain the spin effects due to the Pauli exclusion principle. To this end, we apply a number of SC-IVRs to the scattering of two electrons with initial states corresponding to either parallel or antiparallel spin. We compare the outcome to full quantum results and find that the difference is resolved by those methods that comprise multiple interfering trajectories.

In this work the semiclassical hybrid dynamics is extended in order to be capable of treating open quantum systems considering finite baths. The corresponding phenomena, i.e. decoherence and dissipation, are investigated for various scenarios.

Die Dynamik anfänglich aus dem Gleichgewicht gebrachter wechselwirkender Quantenvielteilchensysteme wirft aktuell noch spannende Fragen auf. In Bezug auf die Thermalisierung ist z.B. nach wie vor ungeklärt, in welcher Form sie überhaupt stattfindet und in welchen Observablen bzw. auf welcher Zeitskala sie zu beobachten ist. Eine ideale Grundlage zur Erforschung von Relaxationsdynamiken in wechselwirkenden Vielteilchensystemen bieten ultrakalte Quantengase aufgrund ihrer guten Kontrollier- und Variierbarkeit. Ein allgemeiner theoretischer Rahmen, auf dessen Basis solche Prozesse zu untersuchen sind, steht jedoch infolge der großen Anzahl der beteiligten Freiheitsgrade bisher nicht zur Verfügung. Für ultrakalte bosonische Gase stellt die Gross-Pitaevskii-Gleichung eines der wichtigsten theoretischen Werkzeuge dar, eine klassische Feldgleichung für die Kondensatwellenfunktion in Molekularfeldnäherung. Die ihr zugrunde liegende Näherung erlaubt jedoch keine nicht-trivialen Aussagen über den vollen N-Teilchenzustand, dessen Kenntnis für die Untersuchung einer möglichen Relaxationsdynamik unabdingbar ist. Um der theoretischen Beschreibung des vollen bosonischen Feldes einen Schritt näher zu kommen, untersucht die vorliegende Arbeit die Anwendung semiklassischer Methoden auf ultrakalte Bosegase. Diese sind in der Regel dann sehr genau, wenn die beteiligten Wirkungen groß gegenüber dem Planckschen Wirkungsquantum sind. Für bosonische Felder wird dieser Grenzfall durch die Bedingung einer großen Teilchenzahl ersetzt. Die immense Anzahl an Teilchen in den hier behandelten Vielteilchensystemen macht die Anwendung semiklassischer Methoden auf diesem Gebiet also vielversprechend. Als zentrales Modellsystem wird ein anfänglich aus dem Gleichgewicht gebrachtes ultrakaltes bosonisches Doppelmuldensystem betrachtet, das eine hochinteressante Dynamik aufweist, die auf das Wechselspiel der Tunneldynamik einerseits und der Wechselwirkung der Teilchen untereinander andererseits zurückzuführen ist. Als Referenz lassen sich aufgrund der speziellen Fallengeometrie im Rahmen der Zwei-Moden-Näherung die Ergebnisse einer numerisch exakten Untersuchung heranziehen. Durch den Einsatz der namhaften WKB-Quantisierung und des besonders aus der Molekülphysik bekannten Reflexionsprinzips wird hier ein geschlossener analytischer Ausdruck für die sogenannte Populationsdifferenz im Doppelminimum hergeleitet, der ausschließlich von den wenigen relevanten Systemparametern abhängt. Diese mächtige Formel erlaubt es nun zum ersten Mal, in quantitativer Weise die charakteristische Sequenz aus Oszillationen, Kollapsen und Revivals in Abhängigkeit der vorausgesetzten Parameter zu untersuchen. Nach dieser ersten erfolgreichen Anwendung semiklassischer Methoden im Modellsystem wird über die reduzierte Dynamik der Populationsdifferenz hinausgegangen. Mithilfe des semiklassischen Herman-Kluk-Propagators lässt sich selbst der volle N-Teilchenzustand untersuchen. Da es letztlich um die Beschreibung ultrakalter Bosonen in beliebigen Potentialen gehen soll, wird zunächst der Herman-Kluk-Propagator für eine Feldtheorie vorgestellt. Im Doppelmuldensystem zeigt sich dann in der Anwendung die semiklassische Propagation in der Lage, für alle untersuchten Parameterregime gute Übereinstimmung mit den numerisch exakten Ergebnissen zu liefern. Zusätzlich findet ein Abgleich der Resultate mit der Truncated Wigner Approximation statt, auf die im Forschungsgebiet ultrakalter Bosonen häufig zurück gegriffen wird. Diese beschreibt die Zeitentwicklung einer Wignerverteilung unter Aussparung der Quanteninterferenzen. In der vorliegenden Arbeit wird gezeigt, dass die Herman-Kluk-Propagation unter Berücksichtigung der Phasen weit über die Truncated Wigner Approximation hinausgeht: Sie gibt alle wichtigen Charakteristika der Dynamik im Doppelmuldensystem wieder. Um die Semiklassik auf ihre Aussagefähigkeit in Bezug auf eine noch komplexere Dynamik zu untersuchen, wird zum Abschluss das Drei-Topf-System betrachtet, das zusätzlich chaotische Regionen im Phasenraum aufweist. Auch hier zeigt sich, dass die semiklassische Berücksichtigung der Phasen die Truncated Wigner Approximation in den Schatten stellt. Allerdings ergeben sich durch die Instabilität der Trajektorien für stark chaotische Regime numerische Probleme, die es in der Zukunft zu lösen gilt. / The dynamics of initially non equilibrium interacting quantum many body systems is an ongoing and interesting field of research. It is still an open question in which form relaxation occurs in such systems, and in which observables and on which timescales a possible thermalization might appear. A perfect playground for the investigations of relaxation dynamics in interacting many body schemes is provided by ultracold quantum gases, which are easily to be controlled and varied in experiments. However, a general theoretical framework for the investigation of such processes is still missing, due to the huge amount of involved degrees of freedom. One of the main theoretical tools in the field of ultracold bosonic gases represents the famous Gross-Pitaevskii equation, a field equation for the Bose-Einstein condensate wave function in terms of a mean-field approximation. However, the underlying approximation prevents the possibility to draw non-trivial conclusions about the full N-particle state, the information of which is necessary for the analysis of relaxation processes. To gain the theoretical description of the full bosonic field, the present thesis deals with the application of semiclassical methods to ultracold boson gases. Those techniques become in general exact, as long as the involved actions are large compared to Planck's constant. For many body systems it turns out that semiclassics are expected to give good results also for the condition of high particle numbers, which is precisely fulfilled in these schemes, making the semiclassical approaches promising. As an essential model system an initially out of equilibrium ultracold bosonic double-well system is investigated. This configuration provides highly interesting dynamics due to the interplay of the tunneling dynamics on the one hand and the interaction amongst the particles on the other. The special trap geometry makes exact numerical calculations in the framework of the two-mode approximation available, which serve in the following as reference data. By applying the common semiclassical WKB approximation and the reflection principle known from molecule physics, a closed analytical expression for the so-called population imbalance of the bosons in the double-well is derived, depending only on the few relevant system parameters. This mighty formula allows for the first time the quantitative investigation of the characteristic sequence consisting of oscillations, collapse and revivals in dependence on the parameters of the system. Since the semiclassical approaches succeeded for the double-well model so far the so-called Herman-Kluk propagator is adopted, to go beyond the reduced dynamics of the population imbalance. The propagator provides the possibility to treat the full N-particle state theoretically and is introduced for the most general case of a bosonic quantum field. Its application to the double-well system yields for all investigated parameter regimes very good agreement with the numerical exact results. Furthermore the outcomes are compared to the Truncated Wigner approximation, which is frequently used in the research field of ultracold bosons. This approach pictures the time evolution of a Wigner distribution, without taking into account the quantum interferences. In the present thesis it is shown that the Herman-Kluk propagation goes clearly beyond the truncated Wigner approach by considering in addition the quantum phases: The propagator is able to reproduce all of the distinctive features of the double-well dynamics. In order to test the performance of semiclassical methods in matters of even more complex systems, the ultracold bosonic triple-well model is finally considered, which exhibits unlike the double-well scheme chaotic regions in phase space. It turns out that the semiclassical propagation outplays again the truncated Wigner approximation. On the other hand the instability of the highly chaotic trajectories causes numerical problems, which have to be solved in the future.

Für das Verständnis einer Vielzahl von Problemen von der Himmelsmechanik bis hin zur Beschreibung von Molekülen spielen Systeme mit mehr als zwei Freiheitsgraden eine entscheidende Rolle. Aufgrund der Dimensionalität gestaltet sich ein Verständnis dieser Systeme jedoch deutlich schwieriger als bei Systemen mit zwei oder weniger Freiheitsgraden. Die vorliegende Arbeit soll zum besseren Verständnis der klassischen und quantenmechanischen Eigenschaften getriebener Systeme mit zwei Freiheitsgraden beitragen. Hierzu werden dreidimensionale Schnitte durch den Phasenraum von 4D Abbildungen betrachtet. Anhand dreier Beispiele, deren Phasenräume zunehmend kompliziert sind, werden diese 3D Schnitte vorgestellt und untersucht. In einer sich anschließenden quantenmechanischen Untersuchung gehen wir auf zwei wichtige Aspekte ein. Zum einen untersuchen wir die quantenmechanischen Signaturen des klassischen "Arnold Webs". Es wird darauf eingegangen, wie die Quantenmechanik dieses Netz im semiklassischen Limes auflösen kann. Darüberhinaus widmen wir uns dem wichtigen Aspekt quantenmechanischer Kopplungen klassisch getrennter Phasenraumgebiete anhand der Untersuchung dynamischer Tunnelraten. Für diese wenden wir sowohl den in der Literatur bekannten "fictitious integrable system approach" als auch die Theorie des resonanz-unterstützen Tunnelns auf 4D Abbildungen an. / Systems with more than two degrees of freedom are of fundamental importance for the understanding of problems ranging from celestial mechanics to molecules. Due to the dimensionality the classical phase-space structure of such systems is more difficult to understand than for systems with two or fewer degrees of freedom. This thesis aims for a better insight into the classical as well as the quantum mechanics of 4D mappings representing driven systems with two degrees of freedom. In order to analyze such systems, we introduce 3D sections through the 4D phase space which reveal the regular and chaotic structures. We introduce these concepts by means of three example mappings of increasing complexity. After a classical analysis the systems are investigated quantum mechanically. We focus especially on two important aspects: First, we address quantum mechanical consequences of the classical Arnold web and demonstrate how quantum mechanics can resolve this web in the semiclassical limit. Second, we investigate the quantum mechanical tunneling couplings between regular and chaotic regions in phase space. We determine regular-to-chaotic tunneling rates numerically and extend the fictitious integrable system approach to higher dimensions for their prediction. Finally, we study resonance-assisted tunneling in 4D maps.

In der vorliegenden Arbeit wird die Fredholm-Methode zur semiklassischen Berechnung einzelner Eigenzustaende von Quantenabbildungen eingesetzt. Es wird gezeigt, wie auch Eigenzustaende zu entarteten Eigenwerten berechnet werden koennen. Die semiklassische Berechnung eines Eigenzustandes erfolgt mittels der Husimifunktion. Es wird gezeigt, wie das Auftreten von Bifurkationen periodischer Bahnen beruecksichtigt werden kann. Dies geschieht auch fuer den Fall von energiegemittelten Eigenzustaenden. Ebenfalls wird die Stoerung einer Quantenabbildung durch einen Punktstreuer und dessen Auswirkungen auf die semiklassische Berechnungen untersucht.