Symmetry effects

In drawing the above conclusions it is important to note that symmetry effects can play a deceptive role if the cavity shape has symmetry. Our example Fig. 3.2a is in the $C_{2v}$ symmetry group, and the symmetry class of the deformations used are given in Table 4.1. In Fig. 4.6 we demonstrate that the addition rule (4.8) is very accurately satisfied at all $\omega$ if $D_1({\mathbf s})$ and $D_2({\mathbf s})$ belong to different symmetry classes of the cavity. Orthogonality of $D_1({\mathbf s})$ and $D_2({\mathbf s})$ is not sufficient to explain this perfect linearity of addition of $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$. Rather, it follows from the symmetry of the kernel $\gamma_{{\mbox{\tiny E}}}({\mathbf s}_1,{\mathbf s}_2)$ of Eq.(3.5). The cross-terms in (3.5) rigorously vanish when such deformations are added, because the kernel must possess the same symmetries are the cavity itself. The consequence is that in order to demonstrate the assertions of this and of the previous section, we had to choose deformations of the same symmetry class, or which break all symmetries of the cavity.

Table 4.1: Categorization of deformations into symmetry classes shared by the ($C_{2V}$-symmetric) example billiard of Fig. 3.2a. See Tables 3.1 and 3.2 for explanation of deformation types.

parity		deformation functions in class
inversion +	no reflections	W$n$ ($n$ even)
	x+, y+	DI, CO
	x$-$, y$-$	RO
inversion $-$	no reflections	W$n$ ($n$ odd)
	x+, y$-$	TY
	x$-$, y+	TX, FR, SX
no symmetries		DF, P1, P2, WG

Figure 4.6: Addition of two `bad' general deformations which come from different symmetry classes of the cavity (1=W2, 2=FR). The two must also be orthogonal, by symmetry. The deviation from linear addition (solid line varying about zero) vanishes at all $\omega$.