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# Decomposition of general deformations

The failure of the WNA for special' deformations also extends to the much wider class of deformations which are similar to special. This is demonstrated in Fig. 4.1. It should be emphasized that this failure happens even if the cavity is strongly chaotic.

We seek an analytical estimate for , and in particular for its zero-frequency limit . This estimate should apply to any (general) deformation, including the case of close-to-special' deformations. It would be useful to regard any general deformation as a combination of special' component and normal' component. The formulation of this idea is the theme of the present section. Supporting numerical evidence is gathered in the next section.

The special deformations (for which we have ) constitute a linear space, meaning that any sum of special deformations is also a special one. Now we would like to conjecture that there is also a linear space of normal' deformations. By definition, for normal' deformation looks like an uncorrelated random sequence of impulses, and consequently the WNA is a reasonable approximation. The notion of randomness can be better formulated as in Appendix F leading to Eq.(F.4). However in practice (F.4) is not useful, because it cannot be applied as an actual classification tool. (Eq.(F.4) is never satisfied exactly). Still we are going to demonstrate that there is a unique way to identify the subspace of normal deformations, if we insist on a maximal (i.e. the most inclusive) definition of this subspace.

It is important to clarify the heuristic reasoning for having a linear space of normal deformations. The that corresponds to some normal deformation looks like white noise. It means that only self-correlations of its spikes are statistically significant. If we have two such generic quantities, say and , then we expect to share the same property.

The correlation function of can be written formally as (4.2)

where is the cross-correlation function. In Appendix E we argue the following  (4.3)

This can also be proved easily using the fact that is an exact quadratic form (3.5) in the function space of . (Consider that the special deformations are eigenvectors of this quadratic form with zero eigenvalue, i.e. they lie in the null-space ). The result is exact, and does not involve any approximation. In Appendix F we argue the following  (4.4)

where . This result is an approximation, which is expected to be as good as our assumption regarding the normality' of the deformation . Consider now the case where is normal and is special. Both Eq.(4.3) and Eq.(4.4) should apply. But these equations are consistent if and only if is orthogonal to . We say that and are orthogonal ( ) using the following definition: (4.5)

Thus we have proved that normal deformations must be orthogonal (in the sense of (4.5)) to special deformations. Obviously we have proved here a necessary rather than a sufficient condition for normality'. However, if we insist on a maximal definition for the subspace of normal deformations, then we get a unique identification. Namely, a deformation is classified as `normal' if it is orthogonal to the subspace of special deformations.

The practical consequences of Eq.(4.3) and Eq.(4.4) are as follows: (4.6)

and  (4.7)

These results are tested in the next section.       Next: Addition of deformations: numerical Up: Chapter 4: Improving upon Previous: Chapter 4: Improving upon
Alex Barnett 2001-10-03