Boundary problems for domains with conical exits at infinity and limit operators

We consider a general boundary value problem (Formula presented.) in a smooth unbounded domain (Formula presented.) with conical exits at infinity, where the coefficients (Formula presented.) belong to the space of infinitely differentiable functions on (Formula presented.) bounded together with all derivatives. We associate with boundary value problem (1) a bounded linear operator (Formula presented.) and we define for (Formula presented.) the family (Formula presented.) of limit operators. We prove that (Formula presented.) is a Fredholm operator if and only if the boundary value problem (1) is elliptic at every point (Formula presented.) and all limit operators (Formula presented.) are invertible.We also consider a realization (Formula presented.) of the differential operator (Formula presented.) as unbounded operator in the Hilbert space (Formula presented.) with domain (Formula presented.) (Formula presented.) We prove that if the boundary value problem (1) is uniformly elliptic in (Formula presented.) then the essential spectrum of (Formula presented.) is the union of the spectra of all limit operators.