Transmission problems for conical and quasi-conical at infinity domains

Let (Formula presented.) be a smooth unbounded domain in (Formula presented.) conical at infinity, (Formula presented.) We consider general transmission problems defined by a differential equation (Formula presented.) and transmission conditions on the boundary (Formula presented.) (Formula presented.) where the coefficients (Formula presented.) are discontinuous on (Formula presented.) functions, such that (Formula presented.) the space of infinitely differentiable functions in (Formula presented.) bounded with all derivatives, (Formula presented.) is a jump of the function (Formula presented.) on (Formula presented.) We give a criterion for the operator (Formula presented.) of the transmission problem (1) and (2) to be Fredholm. We also extend this result to more general quasi-conical at infinity domains. This criterion is applied to the anisotropic acoustic problem (Formula presented.) where (Formula presented.) is a uniformly positive definite matrix on (Formula presented.) with discontinuous on (Formula presented.) entries (Formula presented.) such that (Formula presented.) , (Formula presented.) is discontinuous on (Formula presented.) function such that (Formula presented.) (Formula presented.) is a conormal derivative. We prove that if the acoustic medium is absorbed at infinity the problem (3) has an unique solution (Formula presented.) for every (Formula presented.)