Generalized eigenfunctions of relativistic Schrödinger operators in two dimensions

Generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator H = √ −∆ + V (x) with |V (x)| ≤ C〈x〉−σ, σ > 3/2, are considered. We compute the integral kernels of the boundary values R± 0 (λ) = ( √ −∆− (λ± i0))−1, and prove that the generalized eigenfunctions φ±(x, k) are bounded on R x × {k | a ≤ |k| ≤ b}, where [a, b] ⊂ (0,∞)\σp(H), and σp(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 σ > 2.