It is shown, by means of a simple specific example, that for integrable
systems it is possible to build up approximate eigenfunctions, called {\it
asymptotic eigenfunctions}, which are concentrated as much as one wants to a
classical trajectory and have a lifetime as long as one wants. These states are
directly related to the presence of shell structures in the quantal spectrum of
the system. It is argued that the result can be extended to classically chaotic
system, at least in the asymptotic regime

Baldo, M.

Raciti, F.

English

arXiv

It is shown, by means of a simple specific example, that for integrable systems it is possible to build up approximate eigenfunctions, called {\it asymptotic eigenfunctions}, which are concentrated as much as one wants to a classical trajectory and have a lifetime as long as one wants. These states are directly related to the presence of shell structures in the quantal spectrum of the system. It is argued that the result can be extended to classically chaotic system, at least in the asymptotic regime

Baldo, Marcello

Raciti, F

CERN Document Server

Preprint Dipartimento di Fisica dell'Universita di Catania,
rif. 95/03
Building scars for integrable systems
M. Baldo
1
, F. Raciti
2
1
INFN, Sezione di Catania, 57 Corso Italia , I-95129 Catania, Italy
2
Dipartimento di Fisica and INFN, 57 Corso Italia, I-95129 Catania, Italy
Abstract
It is shown, by means of a simple specic example, that for integrable systems it
is possible to build up approximate eigenfunctions, called asymptotic eigenfunctions,
which are concentrated as much as one wants to a classical trajectory and have a
lifetime as long as one wants. These states are directly related to the presence of
shell structures in the quantal spectrum of the system. It is argued that the result
can be extended to classically chaotic system, at least in the asymptotic regime.
1
1. Introduction
The problem of quantising classically chaotic systems has attracted much attention in the
last few years, especially in view of the subtle connection between classical and quantum
mechanics in the semi-classical limit, generally referred as the h ! 0 limit. Besides the
well studied spectral properties of quantal systems, whose classical counterpart is chaotic,
one of the most interesting and intriguing phenomenon is the appearance of the so called
\scars", i.e. eigenfunctions which display a strong concentration of probability along a
classical closed trajectory. This phenomenon is not yet well understood, and it has been
observed in billiards as well as in smooth hamiltonian systems. It has been argued that
the scars phenomenon can be directly related to the presence of a peak structure in the
spectral density
[1];[2]
, which seems to favour a particular stability of a wave-packet moving
along a classically unstable trajectory. This problem has been considered in ref.[3] where it
has been suggested that scars structures could be revealed through an energy averaging,
in the framework of the Gutzwiller
[4]
formulation of the semi-classical limit. In order
to shade some light on this problem, we consider the problem of establishing to what
extent, for an integrable system, it is possible to construct an (approximate) eigenfuction
which is close as much as possible to a classical trajectory. In the short wavelength limit,
Erhenferst theorem assures that the motion of a wave packet can be as close as one wants
to a classical trajectory, but, of course, an eigenfunction has to be time independent.
Therefore, the motion of the wave packet in the semi-classical limit can only eventually
suggest the particular superposition of wave packets which is (almost) time independent
and still concentrated around the trajectory. However, to estimate the rate of spreading
of a wave packet in the presence of a potential or a barrier is not an easy task, and can be
performed only numerically, even for an integrable system. In the case of an integrable
2
system, one can try to build up directly a superposition of exact eigenstates in order to
construct a wave packet which is concentrated along some classical trajectory and at the
same time survives as long as one can. The essential point of the problem is to recognize
that the lifetime of such a wave packet must be much longer than the characteristic time
of the classical motion, namely the period of the trajectory, otherwise the wave packet
cannot be considered an approximate eigenstate to any respect. This condition is not
fullled in the treatment of ref.[3] for chaotic systems. We will show that in the case of
integrable systems this condition can be fullled to any degree of precision, by building
up a suitable superposition of eigenstates which are approximately degenerate, namely
they belong to the same \shell" of the spectrum. It is hoped that this result could be
of some help of solving the scars problem also in the case of chaotic systems, where the
appearance of a peak in the density of state can be viewed as a sort of an \accidental"
(approximate) shell. Actually, the wave functions belonging to this class, being strongly
localized, are sensitive only to the local shape of the billiard. Therefore, their connection
with classical orbits is not aected if the billiard shape is distorted along the contour
where the wave function is essentially zero. This can include the chaotic cases, at least
for asymptotically large quantum numbers.
2. Method
Let us consider a very simple integrable system with two degrees of freedom, the circular
billiard. The system is trivially solvable, the eigenfunctions are cylindrical Bessel functions
J
l
(
nl
r=R) exp(il), and the 
nl
's are the zeroes of the Bessel fuctions (BF). Here R is
the billiard radius, r the radial coordinate and  the angular coordinate. The quantum
numbers n; l correspond to the quantisation of the radial and angular motion of the particle
3
respectively. This can be seen in the semi-classical limit, namely for large values of the
quantum numbers, for which the asymptotic form of the BF[5] gives the semi-classical
(Bhor-Sommerfeld) quantisation condition
q
k
2
nl
R
2
  l
2
  l
0
= (2n+ 1)=2 + =4 (1)
and it is readly veried that right hand side is just the action integral along the classi-
cal radial motion. For future considerations, it is convinient to keep in mind that the
(constant) angular distance 2
0
between two successive hits of the particle at the billiard
wall is given by cos(
0
) = l=(k
nl
R). Shell structures are associated with closed orbits
6
.
In fact, the condition of k
nl
being stationary for variations n and l of the quantum
numbers, from eq. (1), reads

0

= l
0
=(
n
0
l
0
) =
n
l
=
p
q
(2)
being p and q > p two integer numbers with no common prime factor. This implies that
the corresponding classical trajectory closes afterm = pq hits at the wall, and the quantum
numbers n and l are linearly related. Eq. (2) imposes also a condition for the quantum
numbers l
0
and n
0
, which asymptotically can be satised with arbitrary precision. The
states which have quantum numbers l = l
0
+ l and n = n
0
+ n satisfying the linear
condition of eq. (2) are approximately degenerate, and therefore form an energy shell,
which, in turn, implies the appearance of a sharp peak in the density of states. We want to
show now, by elementary considerations, that for large enough quantum numbers n
0
and
l
0
one can construct linear superpositions of the eigenstates belonging to the same shell,
which satisfy at the same time the two conditions, a) they are concentrated with arbitrary
precision along a classical (closed) trajectory b) they have a lifetime arbitrarly longer than
the classical characteristic time T = M=
nl
h, being M the mass of the particle. Because
4
of the simple symmetry of the system, each eigenfunction has a rotationally invariant
probability density. A wave packet with a narrow angular spread  = 1=
l
can be
written
	
l
0
(r; ) =
X
nl
exp
 
(l   l
0
)
2
2
2
l
!
exp(il)J
l
(k
nl
r) (3)
In order to minimize the energy spread, we restrict the summation to the eigenstates
belonging to the same shell. Asymptotically, for large quantum numbers, the summation
can be performed by expanding the phase of the BF around the chosen value l
0
of the
angular momentum and taking into account the condition of stationary value of k
nl
and
the corresponding linear relation between the quantum numbers n and l. The result reads
	
l
0
(r; )  exp
 
(  (r))
2
2
2
!
(4)
where cos((r)) = l
0
=(k
n
0
l
0
r) , and k
n
0
l
0
= 
n
0
l
0
=R is the eigenmomentum. The wave
packet of eq. (3) is clearly concentrated around the classical trajectory, which is a poly-
gon or a star with m sites, with a spatial spread s  R  R=
l
. The energy
spread E can be estimated in terms of the second derivative k
00
of k
nl
at n
0
l
0
along
the direction dened by eq. (2), E=E = (k
00
=k
nl
)
2
l
. After some algebra, one gets
E=E = g(
l
=
n
0
l
0
)
2
, where g is a costant factor of order unity, and it can be checked
that the higher order terms are vanishing small for asymptotic quantum numbers. The
ratio between the quantal lifetime 
q
= h=E and the classical characteristic time T turns
out

q
T


s
R

2
l
0
(5)
This ratio, for a xed value of the localization s, can be made arbitrarly large by increas-
ing the values of the quantum numbers. Had we chosen a dierent linear combination, on
the contrary this ratio would have been of order unity. In other words, the uncertainity
5
E, with the contraint of eq. (2), is asymptotically of the same order of the mean level
spacing n(E) = 2MR
2
=h
2
. An example, corresponding to p = 1 and q = 3 is depicted
Table 1
l
0
n
0

111 30 241.87
114 29 242.00
117 28 242.09
120 27 242.14
123 26 242.13
126 25 242.07
129 24 241.96
in Fig. 1. The wave function is calculated with the expression of eq. (4) and with the full
expansion of eq. (3) in part a and b respectively. Table 1 reports the quantum numbers
and energies used in the calculation. One can notice that a high degree of localization
can be obtained with only few terms and not too large quantum numbers. In the example

q
=T  15. Higher degree of localization and longer lifetime can be obtained following
the above described procedure.
6
3. Discussion and conclusions
The wave function depicted in Fig. 1 has a striking similarity with some scars reported
in ref. [7] for a chaotic billiard. The authors show that this type of scars \live" in thin
invariant tori embedded in a chaotic region. In the procedure of the present paper the
integrable billiard can be deformed in regions of the contour where the wave function
is vanishing small, and the billiard could become of the chaotic type leaving the scars
essentially untouched. This could be a mechanism of generating scars in a chaotic system.
Of course, the procedure does not exhaust all the possibilities, and indeed in ref. [7] many
examples are shown where the scars \live" in the classically chaotic region (in the Wigner
tranform sense). Another possibility of generalizing the procedure to chaotic billiard is
the case of \local integrability", described in details in ref. [8]. In this case the presence
of an adiabatic barrier allows to expand the hamiltonian, around a closed trajectory, in
the longitudinal and transverse actions, which are approximate constants of the motion
in the vicinity of the trajectory. In this case the present procedure can be repeated step
by step, substituting the angular momentum with the transverse action and the angular
coordinate with the transverse coordinate.
In conclusion we have presented a procedure to build up scars for integrable systems.
Generalizing the method to chaotic systems appears possible, at least when thin invariant
tori exist or local integrability is present. The extension of the method to more generic
cases is under study and the results will be reported elsewhere.
7
References
 [1] Eric.J. Heller,Phys.Rev.Lett. 53 1515 (1994)
 [2] G.G. de Polavieja, F. Borondo, R.M.Benito, Phys.Rev.Lett. 73 1613 (1994)
 [3] E.B.Bogomolny, Physica D 31 169 (1988)
 [4] M.C.Gutzwiller, Chaos in classical and quantum mechanics, Springer-Verlag,
1991
 [5] I.S.Gradshteyn, I.M.Ryzhik, Table of Integrals, series and products, Academic
Press,INC, 1991, formula 8.453
 [6] R.Balian and R.Bloch,Ann.Phys 69 76 1971;
A.Bohr, B.R.Mottelson,Nuclear structure, vol.II, W.A.Benjamin, Inc. (1975), p.
579 and .
 [7] B.Li and M.Robnik, J.Phys A 28 2779 (1995)
 [8] B. Eckhardt et al., Phys.Rev A 39 3776 (1989)
8
Figure captions
 1.a Contour plot of the probability density associated to the wave function (3).
Fixing an angular spread ' = 0:25, and considering the contribution of only 6
values of l centered around l
0
= 120 (cfr. table 1), we have obtained a ratio

q
T
= 15
 1.b Contour plot of the probability density associated to the wave function (4) which
is obtained from (3) expanding the phase of the BF around l
0
taking into account
the condition of stationary value of k
n
l and replacing the summation in (3) by an
integral over l.
9


Building scars for integrable systems

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