GENERALIZED EIGENFUNCTIONS OF RELATIVISTIC SCHRODINGER OPERATORS I

Generalized eigenfunctions of the 3-dimensional relativistic Schrodinger operator root-Delta+V (x) with vertical bar V(x)vertical bar <= C (x)(-sigma), sigma > 1, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernel of (root-Delta-z)(-1) , z is an element of C\[0,+infinity), which has nothing in common with the integral kernel of (-Delta-z)(-1), but the leading term of the integral kernels of the boundary values (root-Delta-lambda -/+ iO)(-1), lambda > 0, turn out to be the same, up to a constant, as the integral kernels of the boundary values (-Delta-lambda -/+ iO)(-1). This fact enables us to show that the asymptotic behavior, as vertical bar x vertical bar -> +infinity, of the generalized eigenfunction of root-Delta + V(x) is equal to the sum of a plane wave and a spherical wave when sigma > 3.