Conclusion

This study is intended to make a contribution to the still developing theory of quantum chaos, by investigating an important model system that in classical mechanics is characterized by weak chaos: the kicked harmonic oscillator.

Well-known classical properties of this model system that also manifest themselves in quantum mechanics -- above all the stochastic webs generated by this system and the diffusive dynamics within the channels of these webs -- make the investigation of its quantum dynamics a nontrivial task.

For the numerical iteration of the quantum map I have chosen to use the matrix representation of the FLOQUET operator with respect to the eigenbasis of the unkicked, i.e. free, harmonic oscillator. While from a numerical point of view it is difficult to evaluate the expressions for the matrix elements of the FLOQUET operator for the kick potential used here, once this has been accomplished one has a very powerful and numerically stable method for iterating the quantum map even for very long times. In practically all cases considered, the numerical error could safely be attributed to the insufficient size $m_{\mbox{\scriptsize max}}$ of the basis used; other sources of numerical error -- which cannot be avoided for finite differences methods, for example -- did not play a major role.

In quantum phase space, the eigenstates $\left\vert m \right>$ of the harmonic oscillator are nicely localized around the corresponding classical trajectories of the harmonic oscillator. This also means that the whole basis $\{\left\vert m \right> \vert \; 0\leq m\leq m_{\mbox{\scriptsize max}}\}$ is localized in a circular region centered about the origin of phase space. Therefore, by systematically increasing $m_{\mbox{\scriptsize max}}$, quantum states in an increasingly large region of phase space can be completely described using this basis. Furthermore, although the $\left\vert m \right>$ are tailor-made for the free harmonic oscillator, they also turn out to be well suited to describe the quantum states that are subject to the nontrivial dynamics of the kicked harmonic oscillator.

The robust numerical method, using this basis with values of $m_{\mbox{\scriptsize max}}$ up to 6000, was fundamental to exploring the quantum dynamics of the model system both for the resonance and nonresonance cases. It allowed to establish the result that, in the first case, the quantum map generates quantum stochastic webs which are very similar to their classical counterparts in several ways. Both types of web are characterized by the same topology and symmetry, and in the channels of both types of web the dynamics is unbounded. While in the classical case the dynamics in the channels is diffusive, the quantum dynamics there is either diffusive or even ballistic. This result is especially interesting as it is in disagreement with the general hypothesis on quantum chaos that classically chaotic systems with diffusive dynamics are characterized by quantum suppression of diffusion -- at least for initial states $\left\vert \psi_0 \right>$ that are located in those regions of phase space (of a weakly chaotic system) being characterized by classically chaotic dynamics.

It has to be mentioned that for obtaining this result the numerical algorithm had to be pushed to its limit, with the consequence of declining accuracy. It would have been desirable to further increase the basis size $m_{\mbox{\scriptsize max}}$ in order to improve on the accuracy and to be able to follow the unbounded dynamics for longer times, but this would have required a considerable additional numerical effort. On the other hand, the result of quantum non-suppression of chaos in the resonance case is supported by a convincing analytical argument, such that its validity can hardly be questioned. However, clarification of this issue might be an appropriate object of future work on the kicked harmonic oscillator.

The numerical results in the case of nonresonance are in agreement with the conventional quantum suppression of diffusion hypothesis, as this is exactly what is found for the model system considered here. This numerical result of quantum localization of the nonresonant kicked harmonic oscillator is given an analytical explanation in terms of the theory of ANDERSON localization. While the general idea of proving localization by mapping the problem onto the well-understood ANDERSON model is motivated by the theory of quantum localization in the kicked rotor, it should be noted that establishing this mapping for the kicked harmonic oscillator was much less straight-forward and required a number of additional considerations. For example, the kicked harmonic oscillator's hopping matrix elements for interaction between the sites of an ANDERSON-like lattice are much more intricate than those of the kicked rotor. Nevertheless, localization could be explained analytically in this way.

Resonances are of particular importance throughout this study. Classically, resonances with respect to $T$ are important because they give rise to classical stochastic webs. The fact that quantum theory predicts the existence of quantum stochastic webs, too, and that these quantum webs are obtained in exactly the same cases of resonance with respect to $T$, is a nice example of quantum-classical correspondence, which has been confirmed numerically in this study. Finally, two theories have been discussed in this study that are entirely independent of each other: the theoretical foundation of quantum stochastic webs and the theory of ANDERSON localization. From a $T$-resonance point of view it is a nice and reassuring feature of the latter that it is consistent with the former in that it predicts localization for all values of $T$ with the exception of the resonant $T$ that give rise to stochastic webs. So both theories combined give the complete quantum picture in a complementary way, and there is no contradiction between the two.

By correspondence, even in the quantum case the resonances with respect to $T$ might be understood to be an essentially classical phenomenon, because they are closely associated with the existence of classical stochastic webs. However, these classical resonances turn out to be true quantum resonances as well, since they are obtained through even two independent truly quantum mechanical theories, without explicitly assuming resonance in the classical case. -- In addition to the resonances with respect to $T$ there exists another class of resonances, with respect to $\hbar$, which is of importance in the analytical discussion of quantum web formation. These $\hbar$-resonances have no obvious classical counterpart, such that they manifest another class of true quantum resonances in the quantum kicked harmonic oscillator.

Finally, I want to point to the fact that the numerical work needed to come to the above conclusions was indeed extensive. First, for investigating the quantum webs above all computers with large memory were needed. Second, in order to study the localization phenomena veritable long-time simulations were performed for which high speed of computation was the prime requirement. Therefore, only the availability of a large number of fast workstations made it possible to study the model system for many different parameter values and initial conditions.

The results obtained here are but a small contribution to the theory of quantum chaos, concentrating on a particular -- though important -- model system. In order to get a better understanding of quantum chaos, more classically chaotic model systems need to be studied with respect to their quantum behaviour. It might be of particular interest to consider other web-generating systems, some of which have been discussed in chapter 1.

For now, a general and comprehensive theory of quantum chaos, applicable to all systems, regardless of the degree of their classical chaoticity, and describing all their essential features, is still lacking.

Wie's dich auch aufzuhorchen treibt,

Das Dunkel, das Rätsel, die Frage bleibt.

Die Frage bleibt

THEODOR FONTANE