Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 143, pp. 1-18.

Title: Generalized eigenfunctions of relativistic Schrodinger operators
       in two dimensions

Authors: Tomio Umeda (Univ. of Hyogo, Himeji, Japan)
         Dabi Wei    (Tokyo Institute of Technology, Tokyo, Japan)

Abstract:
 This article concerns the generalized eigenfunctions of the two-dimensional
 relativistic Schrodinger operator
 $H=\sqrt{-\Delta}+V(x)$ with $|V(x)|\leq C\langle x\rangle^{-\sigma}$,
 $\sigma>3/2$.
 We compute the integral kernels of the boundary values
 $R_0^\pm(\lambda)=(\sqrt{-\Delta}-(\lambda\pm i0))^{-1}$, 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 $\sigma_p(H)$
 is the set of eigenvalues of $H$. 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 $\sigma>2$.
 
Submitted August 19, 2008. Published October 24, 2008.
Math Subject Classifications: 35P10, 81U05, 47A40.
Key Words: Relativistic Schrodinger operators; generalized eigenfunctions;
           pseudo-relativistic Hamiltonians.