  We investigate the algebraic conditions the scattering data of short-range
perturbations of quasi-periodic finite-gap Jacobi operators have to satisfy. As
our main result we provide the Poisson-Jensen-type formula for the transmission
coefficient in terms of Abelian integrals on the underlying hyperelliptic
Riemann surface and give an explicit condition for its single-valuedness. In
addition, we establish trace formulas which relate the scattering data to the
conserved quantities in this case.
