Wave problems in unbounded domains: Fredholmness and the finite section method

We consider a class of strongly elliptic BVP's in an unbounded domain of the form D = {(x, z) is an element of Rn+1 : x is an element of R-n , z > f (x)} where f : R-n -> R is a bounded and uniformly continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and unsteady water wave problems, have been reformulated as second kind integral equations u + Ku = v on BC. Having recourse to the so-called limit operator method, we address two questions for the operator A = I + K under consideration. Firstly, under which conditions is A a Fredholm operator, and, secondly, when is the finite section method applicable to A?.