Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions

Generalized eigenfunctions of the two-dimensional relativistic Schr"odinger operator $H=\sqrt{-\mathrm{\Delta }}+V(x)$ with $\mathrm{\mid }V(x)\mathrm{\mid }\le C<x{>}^{-\sigma }$, $\sigma >3\mathrm{/}2$, are considered. We compute the integral kernels of the boundary values $R_0^{\pm }(\lambda )=(\sqrt{-\mathrm{\Delta }}-(\lambda \pm i0){)}^{-1}$, and prove that the generalized eigenfunctions ${\varphi }^{\pm }(x,k)$ are bounded on $R_x^2\times{k | a\leq |k|\leq b}$,where$[a,b]\subset(0,\infty)\backslash\sigma_p(H)$,and$\sigma_p(H)$isthesetofeigenvaluesof$H$. With this fact and the completeness of the wave operators, we establish the eigenfunction expansion for the absolutely continuous subspace for $H$. Finally, we show that each generalized eigenfunction is asymptotically equal to a sum of a plane wave and a spherical wave under the assumption that $\sigma >2$.