Dispersive treatment of [Formula omitted] and [Formula omitted]

We analyse the rare kaon decays [Formula omitted] and [Formula omitted] [Formula omitted] in a dispersive framework in which the weak Hamiltonian carries momentum. Our analysis extends predictions from lowest order [Formula omitted] chiral perturbation theory ( [Formula omitted]PT [Formula omitted]) to fully account for effects from final-state interactions, and is free from ambiguities associated with extrapolating the kaon off-shell. Given input from [Formula omitted] and [Formula omitted], we solve the once-subtracted dispersion relations numerically to predict the rates for [Formula omitted] and [Formula omitted]. In the leptonic modes, we find sizeable corrections to the [Formula omitted]PT [Formula omitted] predictions for the integrated rates.