A rotationally invariant random interaction ensemble was realized in a single- j fermion model. A statistical approach reveals the random coupling of individual angular momenta as a source for the empirically known dominance of ground states with zero and maximum spin. The interpretation is supported by the structure of the ground state wave functions.

The ∆ 3 (L) statistic characterizes the fluctuations of the number of levels as a function of the length of the spectral interval. It is studied as a possible tool to indicate the regular or chaotic nature of the underlying dynamics, to detect missing levels and the mixing of sequences of levels of different symmetry, particularly in neutron resonance data. The relation between the ensemble average and the average over different fragments of a given realization of spectra is considered.A useful expression for the variance of ∆ 3 (L) which accounts for finite sample size is discussed.

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The ∆ 3 (L) statistic of Random Matrix Theory is defined as the average of a set of random numbers {δ}, derived from a spectrum. The distribution p(δ) of these random numbers is used as the basis of a maximum likelihood method to gauge the fraction x of levels missed in an experimental spectrum. The method is tested on an ensemble of depleted spectra from the gaussian orthogonal ensemble (GOE) , and accurately returned the correct fraction of missed levels. Neutron resonance data and acoustic spectra of an aluminum block were analyzed. All results were compared with an analysis based on an established expression for ∆ 3 (L) for a depleted GOE spectrum. The effects of intruder levels is examined, and seen to be very similar to that of missed levels. Shell model spectra were seen to give the same p(δ) as the GOE. * Electronic address: mulhalld2@scranton.edu