<p>We delineate a class of computationally tractable scattering problems in unbounded domains, which we call <i>decomposable problems.</i> For a decomposable problem, the computational domain can be split into a finite collection of subdomains in which the scatterer has a “simple" structure. A subdomain is simple if the outgoing Green’s function for the operator restricted to this subdomain is either available analytically or can be computed numerically, with arbitrary accuracy, by a tractable numerical method. These subdomain Green’s functions are then used to reformulate the scattering problem as a system of boundary integral equations on the union of the subdomain boundaries. Assuming that the integral equations can be solved using the complex scaling method, this reformulation gives a practical numerical method, to solve these scattering problems, to any desired degree of accuracy, by solving integral equations on <i>finite</i> intervals, using standard discretization techniques.</p>