2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165193We 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.9 pages. Bull. London Math. Soc. (to appear)Exactly Solvable and Integrable SystemsMathematical Physicstext