Numerical comparison of band profiles

The classical band profile (found using methods in Appendix B) and quantum band profile (methods in Appendix C) are compared in Fig. 2.5, for the two-dimensional billiard system shown in Fig. 2.6. This system was chosen because efficient diagonalization methods (see Chapter 6) exist for billiards, and a good basis set (Appendix J) is already known for this shape. The agreement is excellent, well within the expected RMS estimation errors for all $\omega$ considered. Note that there are no fitted parameters in this comparison. The three different choices of the effect of parameter $x$ are different `deformations' of the billiard (the subject of the following two chapters). The range of $\omega$ studied goes from zero to beyond the frequency of the shortest period orbit (the `bouncing ball' orbit family). Note that this agreement has also been tested for other example systems [49,48]. The conclusions can be drawn that classical correlation functions can give a good semiclassical estimate of averaged quantum matrix elements, and QCC in the LRT regime has been tested for (at least some) real chaotic systems.

Figure: Image of the matrix $\partial {\mathcal{H}}/\partial x$ shown as a density plot of $\vert\partial {\mathcal{H}} / \partial x_{nm}\vert^2$, for the case of the `bow' deformation of a quarter stadium billiard. This deformation is not sensitive to `bouncing ball' orbits. Dark pixels correspond to large values. The matrix involves the 422 eigenstates lying in wavenumber range $145 < k < 155$. The band profile gives the average `slice' as shown (also see Fig. 2.3). For $v=1$, the units of $\omega$ and $\kappa$ are the same.

Figure: The same as Fig. 2.7, except for the case of the `bend' deformation, which is sensitive to `bouncing ball' orbits. Notice that there is much more sparse structure. In this and the previous figure, periodic dependence in $E$ can best be viewed by holding up the page and looking along the diagonal.