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\begin{document}
\begin{flushright}
To be published in \\
Mathematical Reviews
\end{flushright}
\beginrev
\reviewer Luca Salasnich
\address
Dipartimento di Matematica Pura ed Applicata
Universit\`a di Padova
Via Belzoni 7
I-35131 Padova, Italy
\author C. Jung and T.H. Seligman
\shorttitle Integrability of the S--Matrix
\journal Phys. Rep. {\bf 285}, 77--141 (1997)
\mcno 1 452 542
\mmrno
\rpclass 81Q50
\rsclass 70K50
\vskip 0.5 truecm
\par
This interesting paper deals with the connection
between the integrability of the
scattering matrix $S$ and the integrability of the Hamiltonian $H$
for classical and quantum Hamiltonian systems.
\par
The authors study the dynamics of a system described
by the Hamiltonian $H$ and for which
the asymptotic dynamics is given by the free
Hamiltonian $H_0$. From these Hamiltonians they derive
the quantum evolution operators $U(t)$ and $U_0(t)$, and also
the corresponding classical flow maps, which are denoted by
$\Phi (t)$ and $\Phi_0(t)$, respectively. Since
the quantum $S$--operator is defined as
$$
S= \lim_{T\to \infty}U_0(T)U(2T)U_0(-T) \;
$$
[see, for instance, J.R. Taylor, {\it Scattering
Theory}, Wiley, New York, 1972], they define a corresponding classical map
$$
{\tilde M}=\lim_{T\to \infty}\Phi_0(T)\Phi (2T)\Phi_0(-T) \; .
$$
Then the authors introduce the classical map $M$, which is the map
${\tilde M}$ reduced to the channel space and for fixed energy.
For a system with $n$ degrees of freedom,
where $M$ has a $(2n-2)$--dimensional domain,
they call $M$ completely integrable if there are $n-1$ independent functions
in involution such that their common level sets are invariant under $M$.
It is shown that integrability of $M$ always leads to integrability
of $H$, but that integrability of $H$ only leads to integrability of $M$
under an additional condition, namely the asymptotic forward and
backward limits must coincide.
\par
In the second part of the paper
the authors transfer their classical results to quantum dynamics.
For a system with $n$ degrees of freedom they call $S$ completely
integrable if there are $n$ independent operators that commute among
themselves and with $S$. Because in quantum mechanics the question of
independence of a set of commuting operators is more subtle than
in classical mechanics, they consider only operators which have a classical
counterpart. In this way it is shown that
integrability of $S$ always implies integrability
of $H$, but integrability of $H$ implies integrability of $S$
under the same additional condition obtained in classical mechanics.
\par
Finally the authors give some numerical example of breaking of
integrability for the $S$--matrix. They show that the nearest neighbour
spacing distribution $P(s)$ of the eigenphases of the $S$--matrix
follows the Poisson distribution
$$
P(s)=exp{(-s)} \; ,
$$
if the matrix is integrable, and that $P(s)$ is very closed to the
Wigner surmise
$$
P(s)=(\pi s/2)\exp{(-\pi s^2/2)} \; ,
$$
if the classical map $M$ is dominated by chaos.
\end{document}