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The discrete spectral problem of Ablowitz-Ladik is considered in the case
in which the potential has a finite support of length L. The spectral
transform is explicitly computed and a recurrence relation on the length L
for computing it in L algebraic step is given. This spectral transform can
be used to generate via the scattering method a finite dimensional version
of the dynamical systems associated to the Ablowitz--Ladik spectral problem.
A special case is considered in which the potential is constraint to evolve
in time on a semi-line. It is shown that the time evolution of the
corresponding spectral data is given by a Riccati equation and that,
consequently, the system is integrable. The truncated soliton, i.e. the
potential obtained by putting to zero the one soliton outside an interval of
length L is examined in detail. The sufficient and necessary condition for
having a soliton contained in the truncated soliton solution is derived.
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