Discussion of the Numerical Methods

Both the TROTTER and the GOLDBERG algorithms do not take into account the FLOQUET nature of the stroboscopically kicked system discussed here. Rather, these algorithms provide quite general methods for numerically treating the SCHRÖDINGER equation and are applicable to a much larger variety of systems. Although being efficient in this general sense -- in particular the GOLDBERG algorithm should be expected to give good results due to the second order CAYLEY approximant (3.18), as opposed to the first order approximation of equation (3.7) of the TROTTER algorithm -- more efficiency can be gained by using methods that take into account the particular properties of the kicked harmonic oscillator. The methods described in sections 3.2 and 3.3 do exactly that, for example by making use of the FLOQUET nature of the system when calculating the propagator for a full period of the excitation.

The finite differences methods are also limited by the disadvantageous feature that there the main computational effort has to be spent for each small time step of length $\delta t$ and must thus be repeated very often, thereby effectively slowing down long-time computations. The methods described in sections 3.2 and 3.3 improve on the finite differences methods by computing the FLOQUET operators once and for all during a more or less lengthy calculation; having accomplished this, application of the FLOQUET operators is then reduced to simple and comparatively fast matrix multiplications.

With respect to the discussion of boundary conditions in subsection 3.1.2, one might at first come to the conclusion that, when studying the quantum analogues of the stochastic webs discussed in chapter 1, the GOLDBERG algorithm with periodic boundary conditions might be the best numerical method to use. But there are several counterarguments to this approach. First of all, the use of periodic boundary conditions automatically incorporates into the numerical algorithm spatial periodicity of the states that are to be computed. If one aims -- as in this study -- at confirming that the investigated system by itself dynamically develops periodic structures in position space (and phase space), then this periodicity should not already be an ingredient of the algorithm. Furthermore, periodic boundary conditions are of no use when studying the quantum analogues of quasiperiodic structures such as the aperiodic webs discussed in section 1.2, or when the case of nonresonance is considered for which no web-like structures should be expected.

On the other hand, once the periodicity of the states with respect to the $x$-coordinate has been established in some way, this observation can be used as a starting point for considerably speeding up the GOLDBERG algorithm, namely by using the smallest possible periodicity interval for $[x_{\mbox{\scriptsize min}},x_{\mbox{\scriptsize max}}]$. In this way the number of nodes needed for representing the wave packet is minimized, thereby minimizing the numerical effort as well. Using periodic boundary conditions also has the advantage that in this way the algorithm avoids cut-off errors altogether that otherwise could arise when the boundary conditions (3.24a) are not satisfied any more, thus spoiling the norm-conservation of the computation.

The problem of cut-off errors does not only occur with respect to the GOLDBERG algorithm, as discussed on page , but with respect to all algorithms discussed here. Localized states -- such as the typical initial states specified in the following chapters -- can be well represented within all the algorithms. But more delocalized nonperiodic states that violate the condition (3.24a) cannot be described and propagated any better within the TROTTER-based algorithm, because the FOURIER transformations used there also require the wave functions (in both the position and momentum representations) to be well localized in the intervals considered. And in the framework of the eigenrepresentation of the harmonic oscillator, a spreading state after a sufficiently long period of time reaches out far enough such that any finite basis $\{\left\vert m \right> \vert \; 0\leq m\leq m_{\mbox{\scriptsize max}}\}$ does not suffice any more to meaningfully expand the state. This observation describes a general and natural restriction for the long-time numerical analysis of unbounded dynamics on a computer. Naturally, this problem does not arise if periodic boundary conditions can be used as discussed above.

If not much computer memory is available, then using one of the finite differences methods might be preferred again: they are characterized by very moderate memory requirements, since both the fast FOURIER transformation and the solving of a tridiagonal linear system need very little memory, as no huge $(j_{\mbox{\scriptsize max}},j_{\mbox{\scriptsize max}})$ matrices need to be stored.

Finally, it is interesting to note the complementary role of the kick propagation within the different numerical methods discussed here. On the one hand there are the methods treated in sections 3.1 and 3.2, where the implementation of the kick is trivial, as these methods are based on the position representation, and the numerically more difficult part is the free harmonic oscillator propagation. On the other hand, the situation is reversed in the framework of the method using the eigenrepresentation of the harmonic oscillator, where the free propagation is easily accomplished, but the kick poses severe numerical difficulties, as discussed in section 3.3.

In the following chapter, the methods described in sections 3.1 and 3.3 are applied to the quantum kicked harmonic oscillator. It is finally established there that using the FLOQUET operator in the harmonic oscillator eigenrepresentation is the most efficient (i.e. fastest) and most general method, allowing to study the quantum analogues of all three interesting classical types of dynamics: periodic webs, aperiodic webs and dynamics in the case of nonresonance (leading to localized quantum motion). Most of the calculations in the following chapters are performed using that method. The position representation FLOQUET operator $U^{\mbox{\scriptsize (pr)}}$ of section 3.2 is not considered any more from here on: it has been shown above to be similar -- in a certain technical sense -- to the $U$ of section 3.3, while needing more computer memory than the latter. But the decisive argument against $U^{\mbox{\scriptsize (pr)}}$ might be that at several points in the following chapters it is just technically more convenient to have the states expanded in terms of harmonic oscillator eigenfunctions.