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.Comment: 14 pages and two figure

Bohigas, O.

Pato, M. P.

Physics Letters B

English

arXiv

AbstractComplete 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

Pato, M.P.

Elsevier - Publisher Connector 

Physics Letters B 595 (2004) 171–176
www.elsevier.com/locate/physletb
Missing levels in correlated spectra
O. Bohigas a, M.P. Pato b
a Laboratoire de Physique Théorique et Modèles Statistiques 1, Bâtiment 100, Université Paris-Sud, 91405 Orsay cedex, France
b Instituto de Fisica, Universidade de Sao Paulo, C.P. 20516, 01498 Sao Paulo, S.P., Brazil
Received 1 March 2004; received in revised form 7 April 2004; accepted 22 May 2004
Available online 22 June 2004
Editor: J.-P. Blaizot
Abstract
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.
 2004 Elsevier B.V. Open access under CC BY license. It is by now well established that, at least with
present statistical significance limited by the rela-
tively scarce amount of high quality data available,
the statistical properties of compound nucleus reso-
nances with fixed quantum numbers are consistent
with the predictions of the Wigner–Dyson random ma-
trix model (Gaussian orthogonal ensemble, GOE) [1].
It is presently understood, not in the form of math-
ematical theorems but rather from numerical experi-
ments and from theoretical insight based on semiclas-
sical approaches, that the origin of this (universal) be-
havior 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 reso-
E-mail address: mpato@if.usp.br (M.P. Pato).
1 Unité de Recherche de l’Université Paris XI associée au CNRS.0370-2693  2004 Elsevier B.V.
doi:10.1016/j.physletb.2004.05.065
Open access under CC BY license. nances of microwave cavities with adequate bound-
aries 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
172 O. Bohigas, M.P. Pato / Physics Letters B 595 (2004) 171–176distort 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 re-
view [7], uses information theory concepts and argues
that correlations in a sequence may provide the nec-
essary redundancy from which error correcting codes
can be constructed. At one extreme where no correla-
tions and therefore no redundancy are present (Pois-
sonian 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 maxi-
mum redundancy and a missed level can be obviously
detected as a hole in the spectrum. Eigenvalues of
random matrices, which exhibit characteristic correla-
tions, correspond to an intermediate situation between
these two extremes. The attempts to locate in the last
case a single missed level have remained unsuccess-
ful so far. However, it should be mentioned that for
two-dimensional chaotic systems where, besides cor-
relations 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 aver-
age spectral density, is well understood. It is then pos-
sible to approximately locate, from the study of spec-
tral 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 sampling). 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 func-
tions Rn(E1,E2, . . . ,En) that give the joint probabil-
ity density of finding levels located around each of the
values E1,E2, . . . ,En. For the incomplete sequence
these functions keep their form and they are only re-
duced by a factor f n since f is the probability of ob-serving one given level. We can therefore write the re-
lation
(1)rn(E1,E2, . . . ,En) = f nRn(E1,E2, . . . ,En),
where small cases denote the actual observed quanti-
ties 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
(2)r1(E) = fR1(E),
and, from (2), the unfolding procedure that maps the
levels onto a new spectrum whose density is equal to
one yields the relation
(3)xk =
Ek∫
−∞
r1(E) dE = fXk,
where k = 1,2, . . . labels the levels which are ob-
served. 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 Yn(X1,X2, . . . ,Xn) from which lower order
correlations are subtracted. From their definition [10]
and (1) we deduce
(4)yn(x1, x2, . . . , xn) = Yn
(
x1
f
,
x2
f
, . . . ,
xn
f
)
.
These important equations show that the correlation
functions keep their form, the only modification is a
rescaling of variables. Thus, if the fraction f is known,
the correlation functions can be reconstructed from the
partially observed spectrum.
It is well known that the number variance Σ2(L)
(variance of the number of levels contained in an in-
terval of length L) and the Dyson–Mehta -statistics
(least square deviation, in an interval of length L,
of the actual staircase counting function from its lin-
ear approximation) are related to the two-point cluster
function through [10]
(5)Σ2(L) = L − 2
L∫
0
(L − x)Y2(x) dx,
O. Bohigas, M.P. Pato / Physics Letters B 595 (2004) 171–176 173(L) = L
15
− 1
15L4
L∫
0
(L − x)3
× (2L2 − 9xL− 3x2)Y2(x) dx.
From (4) it follows immediately that
(6)σ 2(L) = (1 − f )L + f 2Σ2
(
L
f
)
and
(7)δ(L) = (1 − f ) L
15
+ f 2
(
L
f
)
.
Notice the appearance of a linear term in both equa-
tions, if f < 1.
Another set of statistical measures are the E(n, s)
functions that give the probabilities of finding n levels,
with n = 0,1,2, . . . , inside an interval of length s. If
for the complete spectrum they are given by E(n, s),
then their expressions if only a fraction f of levels is
observed is
(8)e(n, s) =
∞∑
k=n
k!
n!(k − n)!f
n(1 − f )k−nE
(
k,
s
f
)
.
This follows from the fact that 1 − f is the proba-
bility that one level is missed, f that it is not and
k!/n!(k − n)! counts the number of ways k − n points
can be removed from k points. From e(n, s), the spac-
ing distributions p(n, s), the probability density of
finding n levels (n = 0,1,2, . . .) between two given
levels separated by a distance s, can be derived using
the relation
(9)P(n, s) = d
2
ds2
n∑
k=0
(n − k + 1)E(k, s);
one obtains
(10)
p(n, s) =
∞∑
k=n
k!
n!(k − n)!f
n(1 − f )k−nP
(
k,
s
f
)
.
For the nearest neighbor distribution (NND), i.e.,
p(s) = p(0, s) (10) becomes
(11)p(s) =
∞∑
k=0
(1 − f )kP
(
k,
s
f
)
.The equations derived above imply that the general
relation
y2(x) = 1 −
∞∑
n=0
p(n,x)
is fulfilled, as it should.
Expression (11) was first proposed as an ansatz
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
(12)E(n, s) = s
n
n! exp(−s)
for the E-functions of a Poisson sequence, one obtains
e(n, s) =
∞∑
k=n
k!
n!(k − n)!f
n(1 − f )k−n
× 1
n!
(
s
f
)n
exp
(
− s
f
)
(13)= E(n, s).
It follows that any other measure derived from the
E(n, s) functions will also remain invariant. Since
Y2(X1,X2, . . . ,Xn) = 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 reads
Y2(x) =
[
sin(πx)
πx
]2
− [Si(πx) − π(x)]
(14)×
[
cos(πx)
πx
− sin(πx)
(πx)2
]
,
where Si(πx) is the sine-integral and (x) = ±1/2 if
x > 0 or x < 0 and (0) = 0 if x = 0. The number
variance Σ2(L) is given by
(15)
Σ2(L) = 2
π2
[
ln(2πL) + γ + 1]− 1
4
+ O(L−1),
174 O. Bohigas, M.P. Pato / Physics Letters B 595 (2004) 171–176where γ = 0.577 . . . is Euler’s constant and the -
statistics is given by
(16)
(L) = 1
π2
[
ln(2πL) + γ + 5
4
]
− 1
8
+O(L−1).
The NND distribution is well approximated by the
Wigner surmise (2 × 2 matrices)
(17)P(s) = P(0, s) = π
2
s exp
(
−π
4
s2
)
.
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 2 × 2 matrix result
(18)P(1, s) = 8
3π3
(
4
3
)5
s4 exp
(
−16s
2
9π
)
(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]
(19)V 2(k)  Σ2(L = k) − 1
6
.
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 Y2,
illustrating the scaling behavior Eq. (4) using (14).
By comparing Eq. (6) with (15) and (7) with
(16) one immediately sees one major effect of the
incompleteness of a sequence. Instead of a logarithmic
increase of the number variance or of the Dyson–
Mehta statistic (f = 1), one has a linear increase for
f < 1 of slope (1 − f ) or (1 − f )/15, respectively.
Consider finally p(s), the NND very often dis-
cussed in the literature. It is given by (11), (17), (18),
(19). The range of the argument determines the num-
ber 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 ana-
lyze high quality data of 238U neutron resonances [13]
and of 48Ti proton resonances [9]. It was concluded
that a fraction of about 10% of the resonances is miss-
ing 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 asymp-
totic behavior of p(s), to rewrite Eq. (11) in a differentFig. 1. The two-level cluster function Y2. Full line: Eqs. (4) and (14).
Points: obtained with numerical simulation for different values of
the fraction f of levels observed.
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 con-
tinuous variable
p(s) =
k=K−1∑
k=0
(1 − f )kP
(
k,
s
f
)
+
∞∫
K
dk√
2πV 2(k)
exp
[
k ln(1 − f )
(20)− 1
2V 2(k)
(
s
f
− k − 1
)2]
.
A steepest descent approximation to the integral can
be worked out by mapping the exponent in the
integrand onto a parabola as
F(k) = −k ln(1 − f ) + 1
2V 2(k)
(
s
f
− k − 1
)2
(21)= F(ks) + t2,
where ks , the stationary point, is the root of the
equation F ′(k) = 0. This transcendent equation can
O. Bohigas, M.P. Pato / Physics Letters B 595 (2004) 171–176 175be solved approximately if one considers the variance
V 2(k) as a slowly varying function of its argument, in
which case one can write
(22)ks = s
f
− 1 + V 2
(
s
f
− 1
)
ln(1 − f ).
Replacing then the coefficient 1/
√
2πV 2(k)(dk/dt)
by its value at the stationary point, we obtain the
closed expression
p(s) =
k=K−1∑
k=0
(1 − f )kP
(
k,
s
f
)
(23)+ 1
2
exp
[−F(ks)][1 − erf(ts)],
where
(24)ts =
{√
F(K) − F(ks), for ks < K,
−√F(K) − F(ks), for ks > K.
In the asymptotic region, s/f large, (23) takes the
simple expression
(25)p(s) ∼ 1
1 − f exp
[
s
f
ln(1 − f )
]
illustrating that the spacing distribution approaches a
Poisson distribution as f tends to zero. We have there-
fore a family of NND distributions, parametrized by
the fraction f of the observed levels, that interpolates
between the GOE and the Poisson statistics.
In Fig. 2 comparison is made with numerical
simulations. It can be seen that for f = 0.90 the
asymptotic regime Eq. (25) is not yet reached in the
range considered whereas for f = 0.50 it is already
reached for s  2.
In conclusion, we have shown how the problem of
missing, at random, a fraction of levels of a correlated
spectrum can be solved. From the observation of an
incomplete spectrum the results presented here have
a twofold application: (i) if the missing fraction is
known, one can recover the statistical properties of
the complete spectrum, (ii) if the statistical nature
of the complete spectrum is known, an estimate
of the fraction of missed levels can be obtained.
This may at first sight appear as too constraining.
Remember, however, that one of the first concerns in
a given experimental setup is to have the efficiency
of the detectors under control, without which little
statistically significant can be extracted. In this respectFig. 2. Nearest neighbor spacing distribution p(s) for two values
of f . Full line: theory Eq. (11) or (23). Dashed line: Wigner surmise
Eq. (17). Dotted line: asymptotic behavior as given by Eq. (25).
Histograms: numerical simulation.
let us emphasize that the linear increase of the number
variance and of the -statistics accompanied by an
exponential decrease of the NND can be a signature of
the incompleteness of the spectrum. It is also possible
in some cases to obtain independent complementary
information. In the case discussed here the analysis of
resonance intensities [9] may provide an independent
information, to be crosschecked with the results of
spectral fluctuations.
Some final remarks are in order: (i) a straightfor-
ward extension of the present Letter can be performed
in order to include effects not of missed but of spu-
rious levels; (ii) in order to extract parameters from
the data, effects due to the finiteness of experimen-
tal samples must be added; (iii) with the advent of the
neutron nTOF facility at CERN [15] a new generation
of measurements of compound nucleus resonances is
likely to take place, for which the methods discussed
here may be particularly adapted; (iv) besides its prac-
176 O. Bohigas, M.P. Pato / Physics Letters B 595 (2004) 171–176tical interest, the one parameter (f ) family interpolat-
ing spectral fluctuations between Wigner–Dyson and
Poissonian correlations is interesting in its own.
Acknowledgements
Fruitful discussions with A. Heine, G.E. Mitchell
and A. Richter are acknowledged. This work is sup-
ported by a project CAPES-COFECUB. M.P.P. is par-
tially supported by the Brazilian agency Conselho Na-
cional de Pesquisas (CNPq).
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Missing levels in correlated spectra 

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Missing levels in correlated spectra

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