Maximum likelihood method to correct for missed levels based on theΔ3(L)statistic

Abstract

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

Spectral statistics of rare-earth nuclei: Investigation of shell model configuration effect

“…The fluctuation properties of quantum systems with underlying classical chaotic behavior and time reversal symmetry correspond with the predictions of the Gaussian orthogonal ensemble (GOE) of random matrix theory. On the contrary, integrable systems lead to level fluctuations that are well described by the Poisson distribution, i.e., levels behave as if they were uncorrelated [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The information on regular and chaotic nuclear motion available from experimental data is rather limited, because the analysis of energy levels requires the knowledge of sufficiently large pure sequences, i.e., consecutive levels sample all with the same quantum numbers (J,π ) in a given nucleus.…”

“…The fluctuation properties of quantum systems with underlying classical chaotic behavior and time reversal symmetry correspond with the predictions of the Gaussian orthogonal ensemble (GOE) of random matrix theory. On the contrary, integrable systems lead to level fluctuations that are well described by the Poisson distribution, i.e., levels behave as if they were uncorrelated [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Open quantum systems and random matrix theory

“…The shape of the ∆ 3 (L) curves for individual opened spectra looked like those of incomplete spectra or ones with intruder levels. In [4] it was seen that the effect on ∆ 3 (L) of intruder levels or missed levels was the same. The problem of spurious levels is addressed in [5] with both ∆ 3 (L) and the thermodynamic internal energy.…”

“…In [4] a maximum likelihood method based on ∆ 3 (L) was developed. The ∆ 3 (L) statistic is the spectral average of the set of random numbers δ i 3 (L).…”

Missing levels in intermediate spectra

“…There has been a line of research that tries to circumvent some of these limitations and even take advantage of them in order to estimate the number of missing levels and the number of mixed symmetries in a particular experimental level sequence [5,[31][32][33][34][35]. These approaches are based on RMT and the key point in order to be able to extract reliable information about missing levels or mixed symmetries is to assume that the spectral statistics coincide with the GOE (or the corresponding RMT ensemble for each symmetry class), that is, the actual experimental spectrum is chaotic.…”