The statistics of spectra

Two assumptions are implicit in this description (i) the levels must be interacting, so that the analogy with classical mechanics, in which the same trajectory fills the whole of phase space, is as complete as possible and (ii) it must be established, at least in principle, that regularity would reappear if the interactions coupling the levels were turned off. 

one is interested in a rather special kind of statistics, viz. the statistics of a dense population of interacting levels. This is the fundamental distinction between chaos and complexity there may arise situations in which levels do not necessarily all interact (they might have different quantum numbers) but are simply present in large numbers, so that their analysis is not possible in practice but could be performed in principle. These are called unresolved transition arrays (UTAs). One can develop [526] a theory of UTAs which yields general theorems about them as a whole. Such theories are a statistical approach to the interpretation of spectra, but are not related to the problem of quantum chaology. 

In the case of strongly interacting levels, it must be the case that the 

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