Generalized eigenfunctions of relativistic Schrodinger operators I

Generalized eigenfunctions of the 3-dimensional relativistic Schrodinger operator $sqrt{-Delta} + V(x)$ with $|V(x)|le C langle x angle^{{-sigma}}$, $sigma$ greater than 1, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernel of $(sqrt{-Delta}-z)^{-1}$, $z in {mathbb C}setminus [0, +infty)$, 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 $(sqrt{-Delta}-lambda mp i0)^{-1}$, $lambda >0$, turn out to be the same, up to a constant, as the integral kernels of the boundary values $({-Delta}-lambda mp i0)^{-1}$. This fact enables us to show that the asymptotic behavior, as $|x| o +infty$, of the generalized eigenfunction of $sqrt{-Delta} + V(x)$ is equal to the sum of a plane wave and a spherical wave when $sigma$ greater than 3.