Abstract: The statistical theory of spectra formulated in terms of random ... Statistical properties of a quantum system were found to be profoundly interrelated.
Nuclear Physics A504 (1989) 562-588 Nosh-Holland, Amsterdam
DYNAMICS
AND STATISTICS
V.V. SOKOLOV Insfifufe
OF UNSTABLE
QUANTUM
STATES
and V.G. ZELEVINSKY
of Nuclear Physics, 630090, Novosibirsk, USSR Received
22 August
I988
The statistical theory of spectra formulated in terms of random matrices is extended to unstable states. The energies and widths of these states are treated as real and imaginary parts of complex eigenvalues for an effective non-hermitian hamiltonian. Eigenvalue statistics are investigated under simple assumptions. If the coupling through common decay channels is weak we obtain a Wigner distribution for the level spacings and a Porter-Thomas one for the widths, with the only exception for spacings less than widths where level repulsion fades out. Meanwhile in the complex energy plane the repulsion of eigenvalues is quadratic in accordance with the T-noninvariant character of decaying systems. In the opposite case of strong coupling with the continuum, k short-lived states are formed (k is the number of open decay channels). These states accumulate almost the whole total width, the rest of the states becoming long-lived. Such a perestroika corresponds to separation of direct processes (a nuclear analogue of Dicke coherent superradiance). At small channel number, Ericson fluctuations of the cross sections are found to be suppressed. The one-channel case is considered in detail. The joint distribution of energies and widths is obtained. The average cross sections and density of unstable states are calculated.
Abstract:
1. Introduction Statistical
methods
are now providing
deep insight into the physics of real quantum
systems. They arise from structure uncertainties instance, disordered solids or atoms in random complexity systems Statistical
of the wave functions.
but, in some conditions, properties
with the dynamical
of a quantum stability
of systems under consideration (for external fields) as well as from the
The latter can take place not only in many-body even for a small number system were found
of phase space trajectories
of degrees
to be profoundly of a corresponding
of freedom. interrelated classical
system. One can schematically subdivide existing statistical approaches ‘) into global and local ones. Theories of the first kind deal with wide energy regions. The main statistical quantity here is the level density p,(E) for given energy E and other exact integrals of motion 1. It results in the coarse description, commonly using thermodynamical concepts. Actually, one should take an ensemble average over typical eigenstates of the given hamiltonian. The Bohr idea of a compound nucleus ‘) is an excellent example. In the local-type approach ‘> a sequence of closeiy spaced levels is considered and the main subjects of interest are correlations and ffuctuations of quantities relating 0375-9474/89/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
V. V. Sokolov,
to specific energy
As for the usual
it is of minor
statistical works
states.
importance
spectroscopy
of Wigner
were begun
V.C. Zelevinsky
dependence
563
of state properties
development
Dyson “) although
long before ‘)_ Usually
and statistics
here. This field can be naturally
“). Its extensive
‘) and
/ Dynamics
empirical
one considers
was initiated studies
on excitation referred
of local
the statistics
to as
by pioneering regularities
of energy
levels and
matrix elements of simple operators “). The progress in this area has provided us with an understanding of the difference between regular and stochastic quantum systems, as connected with quasiperiodic or ergodic behaviour of co~esponding classical systems 9-‘t). The random matrix theory 3,6,‘2) proved
to be an adequate
tool for investigating
local statistics of quantum states. This method implies a higher degree of abstractness in comparison with Gibbs thermodynamics and reveals the most profound properties determined by general features of dynamics (hermiticity, unitarity, 7’-invariance) rather than by specific details. Stochasticity of motion gives rise to the hypothesis of equivalence of bases in the space of states. It means that the distribution function of matrix elements of a hamiltonian is invariant with respect to orthogonal transformations. Thus, one obtains the gaussian orthogonal ensemble (GOE) of random real symmetrical matrices which gives a good description of stochastic motion in actual
systems
(for example,
isolated
neutron
resonances
in nuclei)
as well as in
model calculations. Lifting the T-invariance restriction one comes to the gaussian unitary ensemble (GUE) of complex hermitian matrices. As it is well known, GOE and CUE predict respectively linear and quadratic level repulsion at small spacings. The repulsion reflects the possibility of level crossing only on a manifold of zero measure in the space of random matrix elements. Actually, excited states of physical systems have a finite lifetime. An unstable state is created by an external field and decays afterwards via open channels. Strictly speaking, the above-mentioned standard gaussian ensembles are applicable to discrete stationary levels only. The influence of coupling with the continuum on the properties of internal states has not been su~ciently understood up to now. In fact, such effects have been considered only in some numerical calculations (see e.g. ref. 13)) of nuclear reactions. On physical grounds, one should expect that even in the case of separated resonances (T/L2 % 1) the level statistics will be modified due to instability.
In particular,
one can guess that the level repulsion
at small distances
=zT should be washed out. The remote classical analogue for this phenomenon is originating from non-crossing phase the possibility of intersection of “corridors” space trajectories of conservative motion as a result of dissipation. In the opposite case of overlapped resonances (r/D* 1) the whole pattern is set by the openness of a system, namely by external mixing via continuum 14). This situation is new for statistical spectroscopy. Since unstable states emerge as intermediate ones in various reactions, the complete statistical description calls for extra hypotheses concerning reaction amplitudes. Assuming that the decay of the internal state n into the channel a performs an
V. V. Sokolov,
564
analysis
of a complicated
its component probabilities Thomas
matched within
distribution
V.G. Zelevinsky
wave function to this channel,
/ Dynamics
and extracts
the modulus
one can also obtain
the scope of Wigner-Dyson 15) for neutron
and stabtics
widths
theory. of isolated
squared
the distribution
IAz12 of of decay
In such a way, the Porterresonances
arises
in good
agreement with experiment ‘1. In contrast to this approach, we use below independent statistical assumptions on internal states and decay amplitudes. For separated resonances nevertheless we come to the same Porter-Thomas width distribution. For many years, statistical reaction theory and statistical spectroscopy were being advanced practically independently. direction of a combined description
Only recently the progress was achieved in the of reaction amplitudes and cross section together system 16--‘R).Having in mind primarily investigation
with levels of an open quantum of unstable intermediate states we will pursue
essentially
the same aim of combined
description using methods generalizing those of traditional statistical spectroscopy. Such as approach leads to the problem of spectral properties of nonhermitian random matrices. Ensembles of those matrices provide generalization of gaussian ensembles of hermitian hamiltonians to unstable systems. We proceed from the general theory of resonance nuclear
reactions
“). In good
approximation reaction amplitudes can be represented as sums of pole terms in the complex energy plane. Such poles 8, = E, --iiF,, being the eigenvalues of the non-hermitian effective hamiltonian X correspond to unstable intermediate states with energies E, and widths r,. We adopt simple statistical hypotheses concerning matrix elements of hermitian and antihermitian parts of X. If coupling of internal states with the continuum is weak (small external mixing) and resonances do not overlap, r/D Q 1, our assumptions give the same results theory. The only significant difference is the disappearance small distances
as the Wigner-Dyson of level repulsion at
1Em - E,,] -Cmax {r,,, r,,}.
In the opposite, limiting case of strong external mixing, the results are governed by the algebraic structure of the anti-hermitian part of %. This structure follows, in fact, from the unitarity condition. Therefore its consequences are of a general nature. Due to this structure,
sharp
redist~bution
(“collectivization”)
of widths
occurs
at
r/D 5 1. As a result, k rapidly decaying states are formed (k is the number of open channels) reflecting segregation of fast “direct” processes. The rest of the intermediate states are long-lived and have small excitation cross sections. This discrimination of widths is consistent with the picture 20) of the two-step course of nuclear reactions in the region of overlapping resonances. The similar phenomenon-formation of a short-lived coherently decaying state in a system with almost degenerate levels - is known in quantum optics as Dicke superradiance. Such a state of N identical two-level atoms coupied via the common radiation field has a width r = A+ where y stands for the width of the excited level of a single atom. In sect. 2 we discuss the phenomenological non-he~itian effective hamiltonian X and its connection to the scattering matrix 3 arising from the general reaction theory. In particular, it is shown that the number of nonzero eigenvalues of the
V. V. Sokolov, V.G. Zelevinsky 1 Dynamics and statistics
anti-hermitian different
part
of 2
can not exceed
forms of the secular
sect. 3. In the case of strong state is demonstrated R-matrix
equation external
for one-channel
and compare
different
the number
for complex mixing,
scattering.
k of open
energies
the width
channels.
Two
8, are considered
accumulation
We introduce
representations
565
in
at a single
the phenomenological
of the scattering
matrix.
Sects. 4
and 5 are devoted to discussion of our statistical hypotheses concerning the distributions of matrix elements of hermitian and anti-hermitian parts of X The distributions of corresponding eigenvalues are analyzed. In sects. 4 and 6 the ensemble averages of R- and S-matrices are found as well as the averaged cross sections of scattering and absorption r,, is obtained of complex
processes. The joint distribution function for energies E, and widths in sect. 7 for the one-channel case. In particular, quadratic repulsion
energies
is revealed.
Due to the unitarity
condition,
residues
in the poles
of an amplitude for the one-channel scattering can be expressed in terms of complex energies 8, only. Therefore the statistics of residues follow from the level statistics in this case. Finally, the mean density of unstable states in the complex energy plane is found in sect. 8. Both cases of weak and strong examined.
2. Effective hamiltonian
external
mixing
are explicitly
and scattering matrix
Suppose a complex quantum system (for instance, a highly excited atomic nucleus) near an excitation energy E is considered. Let the given energy region contain N resonance states studied with reactions a + b (a, b = 1,2, . . . , k are open channels). In accordance with ref. 19), the scattering matrix
sha(E) = 8’” - iTb”(E) can be expressed in terms of amplitudes AZ connecting n (n = 1,2,. . . ) N) with a channel state a as follows:
(2.1) an internal
basis
state
(2.2) Here the N x N matrix
in the internal
space
X= H -tiW plays the role of the For simplicity, we do ing contribution can The anti-hermitian
(2.3)
effective hamiltonian; matrices H and W are both hermitian. not take into account pure potential scattering. The correspondbe introduced without difficulties. part W of (2.3) has the form W,,,,=
; A”,A:*, o=,
(2.4)
566
V. V. Sokolov,
V.G. Zelevinsky
where the sum goes over k open channels. (2.1) [ref. ‘“)I. Generally energy dependent.
speaking,
However,
is high in the energy matrix
that reaction
thresholds
Al and
amplitudes
within
elements energy
are negligible
far from this interval).
of S-matrix H,,
are
since the level density
in, the regular
this region
are sufficiently
matrix
is rather smooth,
we are interested
elements
and statistics
Eq. (2.4) assures the unitarity
this dependence
region
hamiltonian
/ Dynamics
variations
(one should Hen&,
of
assume
one can restrict
oneself to considering only the explicit pole energy dependence in (2.2). This means we can use the matrix % as an effective hamiltonian for studying local dynamical and statistical properties of an unstable system. For convenience, we introduce operator notations in the channel space marking N x k matrices
them with a caret. Using rectangular (2.2) and (2.4) in the compact form ~(E)=A.~;_L_A,
Separability
of W causes the important
A = {At} we write down eqs.
W=AA’.
for k < N the matrix
consequence:
W cannot
have more than k nonzero eigenvalues. Indeed, columns of the matrix A are N-dimensional vectors in the inner space. Provided that k < N these vectors span the k-dimensional subspace of the state space. One can choose the remaining N - k basis vectors to be orthogonal to the columns of the matrix A. Then it is clear that the rank of the matrix W is equal to k. It can be shown readily that the nonzero eigenvalues of W coincide with those of the k x k matrix ~=A+A
in the channel
space.
Its elements
X’” =I,,
(2.6) Ai*Az
are scalar
products
of k N-
dimensional vectors A”. For definiteness, we will consider below the T-invariant theory. As it is known, then the choice of channel states is feasible making the S-matrix symmetric. Without loss of generality, make
the inner
the effective
transposed
basis
hamiltonian
matrices).
As a result,
can also be chosen 5Y symmetric,
for the T-invariant
XT= 2
(the
AZ become
the amplitudes
superscipt
real, At = AT.
N eigenvectors
spond
r,!~“”(n = 1,2, . . . , N) of the non-hermitian hamiltonian to exponentially decaying states and the complex eigenvalues 8, = E, -$ir,,
where
(Z,),,
= S,,8,.
Symmetry
!JC&w-
Z! corre-
(2.7)
give their energies E, and widths r,,. With the notation composed of columns (Cl(“),one can write the diagonalizing Z=
case to T marks
)
V for the N x N matrix transformation
of % as (2.8)
of Z leads to pTW=l=C@T.
(2.9)
V. V. Sokolov,
Here second
the first equality
KG. Zelevinsky
expresses
condition.
X is non-hermitian.
of the existence
of common
of the eigenvalue
problem
and statistics
the orthonormalization
one is the completeness
hamiltonian
/ Dynamics
condition
At the same time
Thus eigenstates
decay channels.
567
whereas
!P’!P # 1 since the
$“” are non-orthogonal
Multiplying
the
the equation
by ?P’ from the left and the conjugate
because XV = !PX,
equation
by tY
from the right we get ~+W~d-~~~+~~=+(~-~+)W=-jty’W~=-iA”*A”T. Unlike
A, the tilded
(2.10)
amplitudes i=
A”, = 1 A&b’,“’ n
WTA,
(2.11)
are complex, Ai being the decay amplitude of an unstable eigenstate m into a channel a. Eq. (2.10) is nothing but the matrix form of the Bell-Steinberger relation 22). After diagonalization of resonance terms
of 2, the scattering
matrix
(2.2) takes the form of a sum
ha (2.12)
> Thus,
in adopted
approximation,
reaction
amplitudes
are meromorphic
functions
in the complex energy plane having poles in the points of eigenvalues of X. Residues in the poles are complex. It is essential for the interference of close resonances 23,24).
3. Secular equation and R-matrix. Complex
energies
(2.7) of intermediate
“Collectivization”
of widths
states are roots of the secular
equation
Det(g--X)=0. The structure
(2.4) of the operator
Let us introduce respectively,
the resolvents
G0(8)=(SS-~))‘, In what follows
(3.1)
W guarantees G’(8)
and
non-negativity G(8)
%‘=E-$iT.
G(8)=(%-X))‘,
we will also need the k x k matrix
of the widths
of the matrices
in the channel
I’,,.
H and
SY
(3.2)
space (3.3)
R( ZY)= ATGO( ‘@A
similar to the Wigner R-matrix of the nuclear reaction theory. Matrix elements of R( 8) have poles in the real points 8 = 8, of eigenvalues of the hermitian part H. Eq. (3.1) can be transformed as follows Det(g-X)=Det(SS_)Det[1+fiG”(8)W] = Det (SZ- H) exp {Tr log [ 1 +fiG’(
8)AAT]}.
(3.4)
V. V. Sokolov,
568
V.G. Zelevinsky
/ Dynamics
It is easy to verify that one can permute cyclically result one reduces the secular equation (3.1) to
and stafistics
the matrix
AT under
log. As a
det[l+$R(8)]=0, where
det stands
for the determinant
(3.5)
in the k-dimensional
channel
space.
In the
practically interesting case k < N, the new equation (3.5) is simpler than the original one (3.1). Using the separability of W we get the Dyson-type relation connecting resolvents (3.2) for stable and unstable systems c(g) Insertion
= cO($)-~iG0(8)~[l+~iR($)I-‘~~~0(8).
of eq. (3.6) into eqs. (2.5) and (2.1) gives f(E)
= ATG(E)A
g(E)
In addition,
(3.6)
the useful
formula
=
=
i(E)
1 +$iR(E)
(3.7)
’
1-$6(E) 1 +;iR(E)
(3.8)
’
for the trace of the Green
function
TrG(8)=TrG0(8)+$K(~) follows from (3.6) where the function
(3.9)
K( 8) is given by the trace (Tr) in the channel
space (3.10) As should
be the case, complex
of eq. (3.5). The contributions cancel out.
poles in eqs. (3.6)-(3.10) of real poles
coincide
with the roots
8 = E, from the two terms of eq. (3.9)
In the simplest one-channel case the R-matrix (3.3) turns out to be just a meromorphic function with positive residues (Wigner R-function) and the secular equation (3.5) takes the form l+$R(8)=l+$Tr[G”($)W]=0.
(3.11)
If 8, = E, -@, is a root of eq. (3.11) then 1 -$R( 8*) = 0, i.e. the one-channel S-matrix (3.8) vanishes at 8= 8 :. Taking into account the asymptotic condition lim E+m S(E) = 1 we obtain the factorized representation (3.12) used repeatedly in the literature [see for example ref. 26)]. This representation is equivalent to the sum (2.12) over poles, provided that the complex parameters 8, are roots of the secular equation (3.11). However, the expression (3.12) remains
V.V. Sokolov, KG. Zelevinsky / Dynamics and statistics
unitary
for arbitrary
unspecified
8,. If one takes,
parameters,
cease to be equivalent.
as in ref. 26), energies
then the two representations
En and widths
of the one-channel
The noted fact often slips away. Meanwhile
ation (3.12) with arbitrary
8, may become
incompatible
of W stressed in sect. 2. Such a non-equivalence is especially important ances when the anti-hermitian part W dominates.
569
r,
as
S-matrix
the parametriz-
with the algebraic
structure
in the case of overlapped resonIn the basis of eigenvectors of H
one has (3.13) Accordingly,
the secular
equation
(3.11) can be written
as a pair of coupled
real
equations Y”E” arc arc n (&ET;2+$2=1’ ” (E-E,)2+$2=E*
In the limiting
case of the completely
degenerate
immediately that there exists the unique with nonzero width
eigenstate
operator
H = E * 1, one obtains
of the effective
hamiltonian
r=Cy,=TrW=w n
5%’
(3.15)
accumulated over all individual widths y”. The remaining N - 1 states remain stable. A similar situation ruling out the arbitrary choice of widths r, always occurs if external mixing is strong. Indeed, it follows from (3.14) that if the summarized width w exceeds the energy range AE where levels a, are located, then the broad level with the width ri - w emerges whereas the remaining widths prove to be small. In the limits w/AE
s 1 and N > 1, assuming
the distribution
of E, to be homogeneous
and
symmetric around E = 0, one can obtain easily for the broad level E, = (llW)C”Y&” -O,~,=~[1-O((AE/w)~)].Asforthenarrowlevels(n=2,3,...,N), their energies E,, occur intermittently between neighbouring a, and their widths r,
- (AE/ w)~w/N complement r, to the total value w. For strong external mixing the more convenient form of the secular equation be obtained with the help of the orthogonal transformation diagonalizing
can the
anti-hermitian part W. Since in the one-channel case the matrix W has N - 1 degenerate zero eigenvalues, one can diagonalize in addition the (N - 1) x (N - 1) submatrix of the hermitian part H with the extra orthogonal transformation of the (N - 1)-dimensional subspace. As a result, eq. (3.11) takes the form %-h-lZ($)+$w=O,
(3.16)
where notations
h=H,,,
h, = HI, = Hv,
(3.17)
V.V. Sokolov, V.G. Zelevinsky / Dynamics and statistics
570
are introduced,
E, being
eigenvalues
At h, = 0 only one level is unstable the widths function
through
admixture
R (3.17)
analogous
continuum states. On the basis of the preceding
of the (N - 1)-dimensional having
a width
of the unstable
one. This mixing
to the function paragraph,
submatrix
w. The remaining
R (3.13)
the scattering
is described
describing amplitude
of H.
levels E, get mixing
by the with
is
W
T(E)=Tr[G(E)W]=
(3.18)
E-h-I?(E)+;iw’
Since l?(E) +oo at E + E, the amplitude (3.18) and the cross section zeroes coinciding with E,. Peaks of the cross section are disposed satisfying
have N- 1 at energies
the equation E-h-&E)=O,
(3.19)
which is nothing but the secular equation for eigenvalues of the hermitian part H. The R-function (3.13) has roots of eq. (3.19) as its poles*. Therefore, independently of the power of external mixing, the one-channel cross section has (neglecting the potential contribution) N peaks alternating with N - 1 zeroes**. However, only in the case of weak coupling via the continuum, the cross section peaks lie, according to (3.14) and (2.12), at the points E, = E, of the energies of unstable intermediate states and have near those points the Breit-Wigner shape with widths r,, = -rn. In the opposite case of strong coupling E,, = F,, (v = 2,3,. . . , N; see eq. (7.36) below) so that now the cross section minima rather than its maxima fall at energies of unstable states. The contributions of the broad and narrow resonances happen to be opposite in phase and cancel out. Outside the interval AE occupied by the narrow resonances,
the amplitude
(3.18) has the Breit-Wigner
shape
with the width
w.
One can conclude that the variables h, E,,, h,, and w are more adequate for parametrizing the energy dependence of the one-channel S-matrix then energies E, and widths r,, of intermediate unstable states. The latter are suitable for analysis of the time dependence
of the process.
These two ways of treatment
are complemen-
tary. We would like to note though that the pole parametrization (2.12) is universal for any channel number whereas the simple properties of the expression (3.18) have no analogues in the many-channel case. Thus, the strong coupling with continuum drives creating a short-lived state. Similarly, if there exist decay probabilities, k broad states are formed summed width. This phenomenon observed in
to sharp redistribution of widths k open channels with comparable absorbing practically the whole realistic calculations of specific
* We recall that the energy shifts occurring in the general theory of nuclear reactions are incorporated into H. l* Such a behaviour of the cross section is typical for the one-channel case only. If, apart from the elastic channel, some inelastic channels are opened the elastic cross section has no zeroes. This is the case also for inelastic cross sections at k 2 3 [ref. “)I. The cross section peaks do not coincide, in the general case, with R-matrix poles. These results follow from the many-channel analogue of eq. (3.18).
V. V. Sokolov,
nuclear
reactions
to a compound of separated at N>l Using
13) implies nucleus.
resonances
hearthevalue the term
VG. Zelevinsky
/ Dynamics
the predominance
Quite abrupt
“phase
to development
and statistics
571
of direct processes transition”
of broad
over relaxation
from the uniform
coherently
decaying
pattern
states occurs
w-AE,i.e.(y)=w/N-D=AE/N. “direct
process”
we actually
have
in mind
any
fast process
independently of its mechanism. The duration h/w of this process is essentially shorter not only than the recurrence time h/D but also than the time interval h/AE of fragmentation of the entrance state into more complicated configurations. On the other hand, we have shown that the lifetime of long-lived states is T(w/AE)h/D% factor w/AE. have
started
h/D. This time exceeds the delay time obtained The discrepance is due to the difference of input from the S-matrix
representation
(2.1),
in ref. “) by the assumptions. We
(2.2) based
on the general
reaction theory. In the one-channel case we have obtained from here eq. (3.18) as a consequence. Hence, the only parameter characterizing the instability of intermediate states is Tr W = w. Unlike this, the explicitly unitary form (3.12) of the S-matrix with arbitrary widths was used in ref. *“). As it has been stressed earlier, the two representations are not equivalent. The large lifetime r of compound states as compared with the recurrence time h/D implies that the system reaches thermal equilibrium. Elastic scattering
through
the single-particle
resonance
coupled
with the back-
ground of stable many-particle levels 29) can be considered as a particular example. The single-particle resonance is formed by the optical shell-model potential. The scattering amplitude found in ref. 29) coincides with (3.18), w being the width of the single-particle resonance. Generally, short-lived states are generated by the anti-hermitian part W of the effective hamiltonian. This part forms superpositions of internal states matched to channel wave functions weakly coupled with the internal
ones.
In fact, such a state is identical
to the doorway
state introduced
in
ref. 30). It is clear from (3.18) how the spreading width r’ of the doorway state arises. The influence of the ocean of long-lived states is described by the function k(E). If one replaces the summation over v in eq. (3.17) by integration, this function gets the imaginary
part corresponding
to the width r’ = 27r(ht)/
D - AE. Therefore,
in our case rt - w B r’ - AE. As it has been mentioned
in Introduction,
width collectivization
induced
by strong
coupling via common decay channels (“self-organization” observed in numerical calculations “)) can be considered as a nuclear analogue of Dicke coherent superradiance 2’). Similar effects were found in numerical simulation 3’) of a dissipative spin system.
4. Statistics of the hermitian part of the hamiltonian; In the preceding logical hamiltonian
averaging over GOE
sections we have discussed dynamical properties of a phenomenoX of an unstable system. Now we proceed by treating 2 as a
V. V. Sokolov,
572
member
of an ensemble
postulates
consistent
We consider independent.
V.G. Zelevinsky
of random
with general
the hermitian We suppose
/ Dynamics
matrices.
physical
We assume
N x N matrix
for matrix
H belongs
to the
and their distribution function Such assumptions are known
elements
N’N+‘l/nexp( Here the centre joint distribution
statistical
W parts of X to be statistically
H and anti-hermitian that a real symmetric
distribution
the simplest
principles.
GOE, i.e. matrix elements H,,, are not correlated invariant with respect to orthogonal transformations. to imply the gaussian
and statistics
is ‘)
of H, -$TrH’).
of the energy scale is placed at the origin. function of N eigenvalues .el, . . . , Ed
(4.1)
Eq. (4.1) leads to the
(4.2) The normalization
constant cN=2
in (4.2) is equal 32) to
N(N-1)/4
N(N+l)/4 1 _N ,[!*I.&)]-‘. ( a2 >
As in ref. I*) we have introduced
the scale factor
N in the exponent
(4.3) of eq. (4.1).
As a result, in the limit N -+OO the energy levels E, are located within the Nindependent interval (-a, a) the mean spacing being equal to D = 2a/ N. The essential feature of the GOE is the level correlation described by the preexponent factor in (4.2). The probability for two levels to coincide tends to zero in proportion to their spacing (so-called linear level repulsion). If the T-invariance is not supposed (the GUE case) this factor is replaced by the quadratic one “). Due to orthogonal invariance, the wave functions p”” of stationary
states may
be represented in the GOE by N-component unit vectors {cp’,“‘}spread isotropically over the hypersphere C,,, (cp!,!,‘))‘= 1. In the large-N limit, it gives the gaussian distribution of components cp’,“’with zero mean values and variances ((cpi,“‘)‘) = l/N. Postulating that amplitudes AZ have the same statistical properties as components cp’,“’one easily obtains
the xk-square
distribution
of decay widths where the number
of degrees of freedom k is equal to the channel number (Porter-Thomas distribution for k = 1). This procedure, however, has no serious justification so we prefer to derive the width distribution with the aid of independent statistical hypotheses concerning the anti-hermitian part W (see the next section). Computation of mean values of various quantities over the GOE is simplified in the limit of large N. In particular, in order to average the Green function G’(8) (3.2) one can expand it into a formal series about H and average each item with the use of the pair contraction formula (see eq. (4.1)) (4.4)
V. V. Sokofov, V.G. Zeleuinsky ,I Dynamics and statistics
573
At N + CO,only non-intersecting contractions are surviving 16) since there a trace in the internal space being proportional to N compensates the factor N-’ in (4.4). The summation of those terms gives* 1
(Go(Q)= ~-(u2/4~)Tr(Go(~)}’
(4.5)
From eq. (4.5) we get 33) g’(~)-~(TrG’(g))=~[i(-~~~,
(4.6)
where the sign is determined by the asymptotic condition g”( %‘)= l/ 8 at ‘%: + co. Using expression (4.6) we obtain the mean level density for a stable system p’(E)=
CS(E-E,) (. =
>
=iIm(TrG’(E-i0))
N--$&x2-E2 0(a2- E2)
(4.7)
(the semicricle Wigner rule “)). The analogous GOE - averaging the R-matrix (3.3) leads in the limit N+ co to (R(~P))GoE=(G’(~))~~~‘(~)JZ,
(4.8)
where the matrix ?? has been defined by eq. (2.4). In particular, in the one-channel case
(NWm=g”(~‘)~.
(4.9)
In the similar manner averaging the function I?( ‘8) (3.17), we find
For GOE-averaging the S-matrix connected with the R-matrix by the non-linear relation (3X), we note that at N + a3 contractions (4.4) of random matrix elements belonging to different R-matrices reduce the number of independent traces leading to contributions of higher order in l/N If the channel number k < N then the smallness of l/N can not be recompensed by a trace in the channel space arising when contracting two matrix elements of H from neighbouring R-matrices. Therefore, up to terms of order k/N, 1 -$i,q”( E)??
&%,E = 1 +4ig”(E)2. * This method
is applicable
also to h~miltonians
containing
a regular
(4.11) part along with a random
one.
V. V. Sokoiov, V.G. Zelevinsky / Dynamics and statisiics
574
As one
can
compound the original “direct”
see, the GOE-averaging states
so that the result
matrix
W. In other
has smoothed depends
words,
contributions
only on the matrix
the average
S-matrix
2
of long-lived rather
than
(4.11) describes
on fast
processes.
5. Statistics To investigate
unstable
of the anti-hermitian states of an open
part of the hamiltonian system,
one is in need of information
about Nk real (due to the T-invariance) matrix elements Ax of the total hermitian hamiltonian between internal (n) and channel (a) states as well as about internal matrix elements H,, . We will assume that the statistical properties of these quantities are similar
and can be described
As in (4.4) we have introduced
by gaussian
distributions.
here the scale factor (w)=(Tr
W)=C
l/N
So we suggest
that*
so that the average
va
trace (5.2)
a
is independent of N serving as an invariant characteristic parameter of decay channels under study. As in eq. (4.1), the gaussian distribution of amplitudes AP, follows practically uniquely from the statistical independence and orthogonal invariance in the internal space. Strictly
speaking,
the hypothesis
of statistical
independence
implies
ignoring
the
direct inelastic reactions. However, it is not too difficult to generalize the consideration to include these reactions in our scheme using the method of ref. 34). For simplicity, we suppose open channels to be equiprobable (TV = 7, (w) = kv). Then additional orthogonal invariance in the channel space arises resulting in the joint amplitude distribution function P(Af,
. . .,A:)=(s)““‘*exp(
-Ew)
(5.3)
depending on w = Tr W = C,, (A:)* only. The distribution (5.3) allows us to derive statistical properties of the matrix elements W,,,, (2.4). The diagonal elements W,,, = C, (AZ)2 = ‘y,, are positive, statistically independent and distributed by the Xk-square rule, (5.4) l In the paper by Weidenmiiller I’), it is supposed that the amplitude randomness takes place in the eigenbasis of the hermitian part H only. However, it is difficult to adjust this assumption to orthogonal invariance of the GOE of hermitian parts.
V. V. Sokolov, V.G. Zeievinsky / Dynamics and siatisiics
This derivation
needs
wave functions
(sect. 4). With the channel
tends to the gaussian
no identification
of decay
amplitudes
number
575
with components
increasing,
of
the expression
(5.4)
distribution
ok
-~(Y+l)‘)
= $&JXP{
(5.5)
with the mean value and the variance
(5.6) respectively. These equations were used frequently 35) in order to extract the effective number of open channels from experimental data. Such a procedure can be meaningful only if the is then a weak The trace w its distribution
widths are small and resonances are not overlapping. The matrix W perturbation and the y,,‘s indeed can be treated as resonance widths. of the matrix W is a sum of independent random quantities yn and function (Nk-I)/2 gk(W)
exp
=
(
at Nk s 1 turns
into gaussian
N --w 2rl
(5.7) )
one (5.8)
with the mean value (w) = kv and the small variance
As for off-diagonal 1W,,I
W,,,,, m # n, their
elements
only and can be expressed ~k(Wmn)=L
T(fk)
N
J;;q
(” IW,,,“~)‘kP”‘2 277
(WmJ=O, The analogous
distribution
distribution
function
in terms of the MacDonald
function
Kck-,),2
depends
function (;“I
wm,,~)
(w’,,)=+2=&w)2. of matrix
elements
for a transition
on
K,(Z), ,
(5.10)
(5.11) between
complicated states has been considered in ref. 8). of a certain row or column of W Unlike the original amplitudes AZ, elements are not statistically independent. Variables mutually independent and independent of y,, are the angles f3,, between N k-dimensional vectors A, with components A”, . Using such a representation we have W,j,,=(A,~A,)=Jy,y,cosfL.
(5.12)
576
V. V. Sokolov,
The distribution
of any angle
V.G. Zelevinsky
=’J;;
from the expression
and statistics
em,, is given by the formula
LP’k(e) ensuing
/ Dynamics
(k 2 2)
Qfk) T($(k-1))
for a solid
angle
(5.13)
sink-2e in a k-dimensional
space.
Since a
rotation in the channel space does not change a scalar product (5.12), the total number of random parameters specifying the matrix W is equal to Nk -+k( k - 1) for k s N. For k > N one should add the number i(k - N)(k - N - 1) of rotation generators for the complementary (k - N)-dimensional space getting as a result tN(N + 1) independent of k. The distribution of k non-trivial eigenvalues of the matrix W at k < N is of particular interest. As it has been mentioned in sect. 2 they coincide with k eigenvalues
y* of the matrix
2 = ATA (2.6) in the channel
space.
The distribution sought for can be obtained by the stochastic equation method 36,37). A stochastic process is constructed to describe evolution of the amplitude matrix A in fictitious time r, dA(r)= where the gaussian
random
force is defined
daz=O,
Independently
of initial
-A(T)
conditions,
da: da:
dT+dQ(r),
(5.14)
by
=$ ~“c?~~&,,,, dr.
the distribution
function
(5.15)
of amplitudes
AZ tends
to that of eq. (5.3) at r+ CO. The evolution of eigenvalues ya(r) is determined by (5.14) as well. Diagonalizing the matrix *(r+dr) with the help of perturbation theory with respect to the parameter (dr)1’2 one can get the drift and diffusion coefficients and then write down the Focker-Planck equation for the distribution function of eigenvalues. Its steady solution is (recall that all y’ > 0)
P( y’, . . .
)
yk) =
c,,
(~h/Y~-Yh~)(~YC)‘N-*“‘2exp(
where C, ya =Tr 2 =Tr W = w. The normalization constant methods developed in ref. “) and proves to be equal
-E;
y”),
(5.16)
can be calculated
by
-1
T(k+l)flr(;a)T(;(N-a-l))
I
.
(5.17)
It can be seen from (5.16) that (y”)= 7. When applied to the hermitian part H the stochastic equation method leads naturally to eq. (4.2). The distinction between eqs. (4.2) and (5.16) is caused by the specific form (2.6) of the matrix 2: its matrix elements are finite sums of products
V. V. Sokolov,
of normally
distributed
and the expression
V.G. Zelevinsky
random
variables.
/ Dynamics
and statistics
577
In the limit N + ~0 this difference
-1N 10 x If - yblexp-$z w-T*) 1 I
fades
(5.16) turns into
k(k+l)/4
B(y’,
2”“r(k+l)nr&)
. . .
1 rl
a
n a a outside the domain of compound resonances the two cross sections coincide being equal to (in units of (r/ k2)(2J + 1) where J is the angular momentum and k is the relative initial wave number) a,(E)
= o,.,.(E)
=
=
77* ~ [;(E+JE*-a2)]*++n2 x2
[$(E/a)+JE2/.2-1)]2+,x2’
(El> a.
(6.3)
The parameter x = T/a introduced in (6.3) is a measure of external mixing and resonance overlapping. In the far regions 1E I z=-a, the function (6.3) looks like the wings of a Breit-Wigner resonance with the broad width n (we omit the channel superscript for simplicity).
K V. Sokolov,
578
In
the compound
formation
V.G. Zelevinsky
resonance
/ Dynamics
region IEl< a, the projectile
of an intermediate
compound
system.
$xJlelastic
following
E2/a2+&*+
(6.4)
1)
one
x2
rr,,,.(E)= lwm-112= do not coincide,
can be captured
As a result, the total cross section
x2+xtil-E2/aZ
a,(E)=2(1-ReJS(E)))= and the shape
and statistics
(6.5)
+&E”/a*+$(x’+l)
their difference
rat,,(E) = 1 - kWH/*= being the cross section of absorption accordance with the qualitative analysis
xJ1 -
E*/a*
~XJI-E*/aZ+$(x2+
(6.6)
1)
due to compound nucleus formation. In of ref. *‘), the absorption cross section (6.6)
decreases when the degree of resonance overlap increases. Fig. 1 demonstrates the energy behaviour of the cross sections
(6.4)-(6.6).
shape
form symmetric
-
3.5
elastic
-3.0
and the total cross section
-2.5
-2.0
-+.5
f
-1.0
have the double-humped
The
1 I
-0.5
Cl
0.5
i.0
1.5
2.0
2.5
30
35
E/o
Fig. 1. Energy behaviour of average cross sections; solid lines 1 and 2 correspond to the shape elastic cross section and to the absorption cross section, respectively; the dashed line 3 corresponds to the total .cross section.
V. V. Sokolov,
V.G. Zelevinsky
with respect to the centre of a spectrum. the minimum
is in the centre.
The maxima
The relative
(+,(O) x2+1 _==~l-g,(a) x(x + 1)
/ Dynamics
drops
and starisiics
579
lie at the edges E = *a whereas
of the minima
1
us..%
x’
a,.,.(a) =(X+1)Z=
are
x2+ 1
2
(6.7)
l-2
The approximate equalities correspond to the strong mixing limit when the middle parts of resonance curves flatten. This flattening is the manifestation of the spreading width (see sect. 1).
7. Distribution of complex energies for an N-level one-channel system As we have noted in sect. 5, at k < N the matrix
W is specified
by Nk - qk( k - 1)
random parameters. Recollecting N eigenvalues of the hermitian part H we have altogether I= N( k + 1) -$k( k - 1) parameters on which the N complex energies are dependent. Being interested in the distribution of energies and widths we have to integrate over I-2N = (k - l)( N -ik) extra variables. In the exceptional case of k = 1 the number
1 of original
variables is just equal to the number 2 N of parameters of integration drops out and the problem becomes transformation and calculation of the jacobian. This one-channel
{E,,, T,,} so that the necessity
reduced
to variable
case will be considered below in detail. We use the secular equation (3.11) with the R-matrix with respect to levels and perform the transformation define characteristic polynomials of N x N matrices
in the form (3.3) symmetric {E,, -y,,}+ {E,,, I’,,}. Let us H, X = H -$W and Xt =
H+iiW:
Qo(h)=Det(A-H)=n(h-E,,);
(7.1)
n Q(h)=Det(A-X)=n(A-g,,), n Q(A)=Det(A-X+)=fl(A-%z).
(7.2)
” One can easily verify that due to eq. (3.13)
aQo(A 1
Q(A)=QdA)-$iC %,a~. m
The coefficients
AiN,” of polynomia Qo(A)=
Q(A)=
(7.3)
m
1s (7.1) and (7.2) defined
according
to
; (-l)“A’,N’(~)AN-n, n=O
; (-l)“A’,“‘(8)AN-“, n=O
Q(A)=
; II=0
(-l)“A$NN’(8*)AN-”
(7.4)
V. V. Sokolov,
580
are symmetric
V.G.
Zelevinsky
over their N variables
AhN’(&)= 1 )
AiN’
/ Dynamics
and linear
of coefficients
in each of them: A:N’(E) = 1 EmEn,. . . ) f?l ’ “lJ2J(E,-EJ2+:(r,+rJ2=
(7.30)
2
( > c
n
After integrating resonances 9v2,.
rn
(7.31)
-,r~=2r,~2r~+[(~~2r;)‘-~2r:].
over E, and r, one obtains
the distribution
function
for long-lived
(m, n = 2,3, . . . , N): . . , EN;
r2v..
. , rN)
T(N-4)
[I + Wd
c, miN-1/2
)I .
x exp
(7.32)
To estimate values of typical small widths it is convenient to introduce amplitudes Pn = a (n 3 2) assuming that they run over the whole real axis. Then dr, ;z+rJdB” where
G =C,“=, r,
dimensional density
= G(N-3)‘2 dG do,_,
and d ON-1 stands
for an infinitesimal
space with Pn as its coordinates.
(7.32) has a maximum
(7.33) solid angle of a (N - l)-
It is easy to show that the probability
at (7.34)
whereas the angle distribution in the space of Pn is nearly isotropic. Hence, the individual small widths are of order rnz2 - G,/( N - l), i.e., at large N, r,,, - n/z2N and the corresponding overlap parameter is r/D - (T/x2N)N/a = l/x < 1. As a consequence, long-lived states fall again in the small overlap regime, and, at spacings S,, < l/x, the energy distribution coincides with that of eq. (4.2).
V. V. Sokolov,
584
These results
are evidently
they can be derived The version
(3.16) obtained
study,
/ Dynamics
of dynamical
from the secular
for the case x * 1. Since, limit under
V.G. Zelevinsky
rather
equation
and statistics
than statistical
by diagonalization
of the matrix
due to eq. (4.4), (ht) = a2/4N
we can use the perturbational
%,=h-$w+
c”h -E, ht-$iw
origin.
Therefore
(sect. 3) straightforwardly. W is appropriate
= n2/4x2N
is small in the
formulae:
=h-$w+I?(h-fw),
(7.35)
“-2
h:
q,=&“+
Neglecting (h - .z~)~c u2 as compared (7.36) the large width
with
N
r,=w-w
vs2.
E,-h+$w’
(7.36)
w2= n2 we find from eqs. (7.35) and
ht
(7.37)
U;2(h_E,)2++W2=
as well as small ones (7.38) A similar picture has been discussed by Shapiro 2”) based on quite different arguments. In contrast to that paper, we are not in need of any assumptions concerning the preexistence of narrow resonances on the background of a broad one. The separation of width scales happens automatically by virtue of the structure of the matrix W which is dictated ultimately by the unitarity principle. It should be stressed that in the one-channel case the width rearrangement leaves no room for manifesting Ericson fluctuations 38). Indeed, at N> 1, the conditions necessary for the strong overlap of resonances cannot be implemented. To reveal such fluctuations
there
should
be k B 1 open
channels
when
k short-lived
states
with comparable widths and weakly correlated decay amplitudes arise. To investigate rigorously the cross section fluctuations in the framework of our approach, one should derive the distribution problem elsewhere.
of decay amplitudes
8. Average
density
An important quantity for an N-level complex energies %,, = En -$ir, (compare
(2.11). We hope to return to this
of unstable
unstable (4.7)),
states
system
= C6(E-E,)6(r-I’,) n >(
is the mean
density
of
(8.1)
V. V. Sokolov, V.G. Zeleoinsky / Dynamics and statistics
normalized
according
to dTp(E,
At x < 1 the anti-hermitian order
585
part
8, = E, - $y, . The density
r) = N.
(8.2)
W is a small perturbation
so that in the lowest
p (E, ZJ is then factorized,
P(E, r) = p’(E)%(r)
(8.3)
where p”(E) is the semicircular distribution (4.7) and C?‘,(r) stands Thomas distribution (5.4) with the mean value (r) = n/N.
for the Porter-
In the opposite case of x > 1, it is convenient to use the alternative version (3.16) of the secular equation. Eigenvalues of X can be found by perturbation theory (7.35), (7.36). Separating in eq. (8.1) the short-lived ones (n=~=2,3,..., N) we have
(n = 1) states from long-lived
p( E, ZJ = ;(Sc2’( ‘2’)- 8,)) +; C (S(*)( i!T- Z$,)) . Averaging over h, and E, in the first term (denoted at N + cc like in (4.9) resulting in
(8.4)
by p,( E, I’)) can be carried
pl(E,~)=~(6(2)[~-h+~iw-$a2g0(h-~iw)])h,W. Quantities
h and w are normally
distributed
out
(8.5)
with mean values
(h) = 0 and (w) = n
and variances (h2)= n2/2x2N and ((~-77)~)=277~/N so that we can expand the argument of the S-function in eq. (8.5) over h/w. In the real part, the corrections cancel out up to the second order inclusively result coincides with that of eq. (7.37). Hence,
whereas in the imaginary part the we obtain in this approximation
p,(EJ)=(cT(E-~)~(CW+W/X*)),,,,. Final
averaging
over h and w can be performed
(8.6) trivially
giving
N[C~(1-x-2)]2 v2( 1 - X-*)2
.
(8.7)
Within our accuracy, the energy and the width of the short-lived state are not correlated. This resonance is located near the centre of the region of levels coupled with a given channel. The whole energy region is covered by the resonance width. As for fluctuations of E and r they are relatively small, -l/m. The second term of eq. (8.4), p,(E, r), contains N - 1 identical contributions. Each of them is equal to
(8.8)
V. V. Sokolov,
586
V.G. Zelevinsky
/ Dynamics
and statistics
with notations I=
wE+hT
Now
it is obvious
f=
’
w+r
wr [a+(g’].
(8.9)
that (see eq. (4.7)) (8.10)
(8.11) The remaining The result is
averaging
over h and w can be done
p2(E,r)=(N-I)~&I~-E~B(u~-E~)
by the saddle-point
method.
(-gr) .
Jsexp
(8.12)
We keep here unity side by side with N to guarantee proper normalization. Thus, for N - 1 long-lived states the semicricle GOE rule (4.7) along with the PorterThomas distribution (7.38), (r) = 7/x2N.
are recovered. However, the latter has a small average width As we have emphasized earlier, overlap of long-lived resonances
is small, (r)/D - l/x. In the general case of arbitrary external mixing one can use the electrostatic analogy to reduce the calculation of the average density (8.1) to the more convenient averaging of the trace of the Green function (see ref. 39)), (8.13) The function (8.13) acquires the meaning of the two-dimensional electric field created by the charge distribution p( E, r). Then, by virtue of the Maxwell equation, (8.14) It allows to perform
the explicit
averaging
A,. The remaining
GOE-averaging
over the gaussian
is still a complicated
ensemble
of amplitudes
task.
9. Conclusion In this paper we have attempted to extend standard statistical spectroscopy of discrete levels to unstable states. Decaying systems are described by ensembles of random non-hermitian hamiltonians represented by N-dimensional matrices. Complex eigenvalues of these matrices determine energies E, and widths r, of states with a finite lifetime. Under simple statistical assumptions corresponding to completely stochastical motion, the eigenvalue distribution in the complex plane (E, r) is obtained. The one-channel case is discussed in detail.
V. V. Sokolov, KG. Zelevinsky / Dynamics and statistics
The instability repulsion
at distances
they repulse increasing
of states changes smaller
quadratically coupling
place. When matrix
significantly
than widths.
with respect
with the continuum, elements
the energy statistics,
As for complex
to the distance dynamical
of the anti-hermitian
587
removing
energies
8, = E, -$I’”
in the complex
restructuring
level
plane*.
At
of a system takes
part of an effective hamiltonian
become comparable with the spacings of eigenvalues of the hermitian part, a transition to a new regime occurs. In the new regime the picture is determined by the algebraic structure of the anti-hermitian part dictated by unitarity rather than by particular statistical hypotheses. As a result, decay probabilities turn out to be redistributed in such a way as to distinguish short-lived states (their number is equal to that of open decay channels) saturating approximately the whole summed width of levels. The remaining long-lived states are quite isolated and their statistics are essentially the same as those of stable states. Such a picture implies the coexistence of direct and compound processes resulting in a gross-structure of the cross section. A similar structure with several short-lived resonance states has been observed in numerical simulations of nuclear reactions 13). In this work formation of such a structure along with the state overlap increasing was referred to as “self-organization”. As one can see its mechanism is of a quite general to Dicke superradiance in quantum optics.
nature,
being analogous
In the one-channel case there is no room for manifestation of Ericson fluctuations. In principle, they might be observed if the channel number is large enough to generate a number of interfering resonances. This problem is under study. The main results of this paper have been published in ref. 4’). We thank Ingrid Rotter for stimulating discussions and making information on her results available to us prior to publication. We appreciate useful discussions with M.G. Urin;
we are grateful
also to F.M. Izrailev
for permanent
interest
to the
work and to L.F. Khailo for help with the calculations. In addition, one of us (V.S.) would like to mention fruitful discussions with V.L. Lyuboshitz, and the other (V.Z.), those with I.V. Krivoshey.
References 1) 2) 3) 4) 5) 6) 7) 8)
A. Bohr and B. Mottelson, Nuclear structure, vol. 1 (Benjamin, Massachusetts, 1969) N. Bohr, Nature 137 (1936) 344 Statistical theories of spectra: Fluctuations, ed. C.E. Porter (Academic Press, New York, S.T. Belyaev and V.G. Zelevinsky, Sov. Phys. Usp. 28 (1985) 853 E.P. Wigner, Ann. Math. 53 (1951) 36; 62 (1955) 248 F.J. Dyson, J. Math. Phys. 3 (1962) 140, 157, 166 1.1. Gurevich, Zh. Eksp. Teor. Fiz. 9 (1939) 1283 T.A. Brody et al., Rev. Mod. Phys. 53 (1981) 385
* This is equivalent to cubic repulsion if one uses polar coordinates in the (E, r)-plane. is confirmed by the numerical calculations “) for a kicked nonlinear top with dissipation.
1965)
Such a rule
588 9) 10) 11) 12) 13) 14)
15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41)
V. V. Sokolov,
V.G. Zelevinsky
/ Dynamics
and statistics
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