Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions

Abstract

Generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator $H=\sqrt{-Δ}+V(x)$ with $|V(x)|\leq C< x>^{-σ}$, $σ>3/2$, are considered. We compute the integral kernels of the boundary values $R_0^\pm(λ)=(\sqrt{-Δ}-(λ\pm i0))^{-1}$, and prove that the generalized eigenfunctions $ϕ^\pm(x,k)$ are bounded on $R_x^2\times\{k | a\leq |k|\leq b\}$, where $[a,b]\subset(0,\infty)\backslashσ_p(H)$, and $σ_p(H)$ 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 $σ>2$.