  In this work we discuss in detail the non-perturbative determination of the
momentum dependence of the form factors entering in semileptonic decays using
unitarity and analyticity constraints. The method contains several new elements
with respect to previous proposals and allows to extract, using suitable
two-point functions computed non-perturbatively, the form factors at low
momentum transfer $q^2$ from those computed explicitly on the lattice at large
$q^2$, without any assumption about their $q^2$-dependence. The approach will
be very useful for exclusive semileptonic $B$-meson decays, where the direct
calculation of the form factors at low $q^2$ is particularly difficult due to
large statistical fluctuations and discretisation effects. As a testing ground
we apply our approach to the semileptonic $D \to K \ell \nu_\ell$ decay, where
we can compare the results of the unitarity approach to the explicit direct
lattice calculation of the form factors in the full $q^2$-range. We show that
the method is very effective and that it allows to compute the form factors
with rather good precision.
