Missing levels in correlated spectra

Complete spectroscopy (measurements of a complete sequence of consecutive levels) is often considered as a prerequisite to extract fluctuation properties of spectra. It is shown how this goal can be achieved even if only a fraction of levels are observed. The case of levels behaving as eigenvalues of random matrices, of current interest in nuclear physics, is worked out in detail.

It is by now well established that, at least with present statistical significance limited by the relatively scarce amount of high quality data available, the statistical properties of compound nucleus resonances with fixed quantum numbers are consistent with the predictions of the Wigner-Dyson random matrix model (Gaussian Orthogonal Ensemble, GOE) [1]. It is presently understood, not in the form of mathematical theorems but rather from numerical experiments and from theoretical insight based on semiclassical approaches, that the origin of this (universal) behavior has to be found in the chaotic character of the underlying classical motion [2]. Analyses have been mainly performed for two degrees of freedom systems (one particle in a two-dimensional box), which can be also experimentally exemplified in studying resonances of microwave cavities with adequate boundaries or of quartz or metallic blocks (elastodynamics) (see, for instance, contributions by A. Richter, H. J. Stöckmann and C. Ellegaard [3]).
Efforts have also been devoted to find out from which excitation energy on, starting from the ground state (where, by analyzing the systematics of ground state energies, evidence for the coexistence between regular and chaotic motion has been given [4]) the 'random matrix theory' behavior holds [5]. This is a difficult task because, below neutron threshold, one can hardly be sure that all levels are detected. Work is also currently performed to investigate the statistical properties of the fine structure of analog or other giant resonant states, at excitation energies which may be below or far above the neutron threshold [6].
At different degrees, most of these studies must face the difficulty of establishing the extent to which the sequences of energies analyzed are complete ( no missing levels) as well as how the missing levels may distort the statistical properties of the otherwise ideal (complete) sequence. It is the purpose of this letter to present some results which may be useful in this context.
The most ambitious goal may be stated as the one of detecting the location of, say, one missing level on an otherwise complete sequence. Dyson, in a recent review [7], uses information theory concepts and argues that correlations in a sequence may provide the necessary redundancy from which error correcting codes can be constructed. At one extreme where no correlations and therefore no redundancy are present (Poissonian sequence), there is no possibility of detecting one missing level. At the other extreme, a sequence of equally spaced levels (picket fence ), there is a maximum redundancy and a missed level can be obviously detected as a hole in the spectrum.
Eigenvalues of random matrices, which exhibit characteristic correlations, correspond to an intermediate situation between these two extremes. The attempts to locate in the last case a single missed level have remained unsuccessful so far. However it should be mentioned that for two-dimensional chaotic systems where, besides correlations of the order of one mean spacing as described by random matrices, the presence and the role of long range correlations governed by the shortest periodic orbits and reflected in Weyl's law describing the average spectral density, is well understood. It is then possible to approximately locate, from the study of spectral fluctuations, a single missed level [8].
Here we address a less ambitious question, namely to study how the spectral fluctuations of an ideally complete sequence are affected when a fraction f (0 < f < 1) of levels are detected. It will be assumed that the sequence is infinite and stationary and that the levels are dropped at random from the complete sequence (random sam-pling). The last assumption is presumably justified when observing compound nucleus resonances with limited detection sensitivity. Indeed, in random matrix theory level positions and widths are uncorrelated, a feature which is confirmed by the analysis of experimental data.
We start considering the n-point correlation functions R n (E 1 , E 2 , ..., E n ) that give the joint probability density of finding levels located around each of the values E 1 , E 2 , ..., E n .
For the incomplete sequence these functions keep their form and they are only reduced by a factor f n since f is the probability of observing one given level. We can therefore write the relation where small cases denote the actual observed quantities and capital letters those of the complete spectrum. We shall keep this notation in what follows, namely use the same letter, capital or small case, to refer to corresponding quantities for complete or incomplete sequences, respectively. For the spectral density (1) gives and, from (2), the unfolding procedure that maps the levels onto a new spectrum whose density is equal to one yields the relation where k = 1, 2, ... labels the levels which are observed. The normalization will always be kept the same (the mean spacing < s >= 1 for the complete as for incomplete sequences). This is in contrast to some of the expressions given in [9].
One is usually interested in the n-point cluster functions Y n (X 1 , X 2 , ..., X n ) from which lower order correlations are subtracted. From their definition [10] and (1) we

From (4) it follows immediately that
and Notice the appearance of a linear term in both equations, if f < 1.
This follows from the fact that 1 − f is the probability that one level is missed, f that it is not and one obtains For the nearest neighbor distribution (NND), i.e. p(s) = p(0, s) (10) becomes The equations derived above imply that the general relation is fulfilled, as it should.
Expression (11) was first proposed as an anzatz in ref. [11], and in ref. [9] it has been shown that the coefficients f (1 − f ) k maximize the Shannon entropy with constraints appropriately defined. Here it is obtained in a direct way.
We first apply, as a check, the previous equations to a Poissonian spectrum of uncorrelated levels in which case they must remain invariant (they should not depend on f ). If, for example, we substitute in Eq. (8) the expression for the E-functions of a Poisson sequence, one obtains It follows that any other measure derived from the E(n, s) functions will also remain invariant. Since Y 2 (X 1 , X 2 , ..., X n ) = 0, the n-point cluster functions of the incomplete sequence also vanish and all other measures derived from them also remain invariant.
Before considering GOE spectra with a fraction of observed levels, let us first recall some of the GOE expressions (f = 1) to be used [10]. The two-point cluster function where Si (πx) is the sine-integral and ǫ (x) = ± 1 2 if x > 0 or x < 0 and ǫ (0) = 0 if x = 0. The number variance Σ 2 (L) is given by where γ = 0.577... is Euler's constant and the ∆-statistics is given by The NND distribution is well approximated by the Wigner surmise (2x2 matrices) The next to nearest neighbor distribution P (1, s) is given by the NND of the symplectic ensemble which again is well approximated by the 2x2 matrix result (care has been take to insure s = 2). The higher (k = 2, 3, ...) spacing distributions P (k, s) are well approximated by their (Gaussian) asymptotic form, centered at k + 1 and variances V (k) given by [12] We apply now the above expressions to GOE spectra when only a fraction f of levels is observed. On Fig 1 is displayed for different values of f the function Y 2 , illustrating the scaling behavior Eq. (4) using (14).
By comparing Eq. (6) with (15)and (7) with (16)  Consider finally p (s), the NND very often discussed in the literature. It is given by (11), (17), (18), (19). The range of the argument determines the number of terms to be included in (11). The slope at the origin given by the first term in (11) is increased by a factor 1/f with respect to the GOE value. Let us also mention that, in fact, Eq. (11) was already used to analyze high quality data of 238 U neutron resonances [13] and of 48 Ti proton resonances [9]. It was concluded that a fraction of about 10% of the resonances is missing in both cases and that the correction is important in investigating the parity dependence of nuclear level densities [14].
It is convenient, in particular to exhibit the asymptotic behavior of p (s), to rewrite Eq. (11) in a different form. Let us first separate the first K terms in the sum and approximate the rest (the infinite sum from k = K to k = ∞) by an integral in which k is treated as a continuous variable A steepest descent approximation to the integral can be worked out by mapping the exponent in the integrand onto a parabola as where k s , the stationary point, is the root of the equation F ′ (k) = 0. This transcendent equation can be solved approximately if one considers the variance V 2 (k) as a slowly varying function of its argument, in which case one can write Replacing then the coefficient ( where In the asymptotic region, s f large, (23) takes the simple expression illustrating that the spacing distribution approaches a Poisson distribution as f tends to zero. We have therefore a family of NND distributions, parametrized by the fraction f of the observed levels, that interpolates between the GOE and the Poisson statistics. In