Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions

Abstract

Generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator $H=\sqrt{-\Delta }+V(x)$ with $|V(x)|\le C<x{>}^{-\sigma }$, $\sigma >3/2$, are considered. 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 $R_x^2\times \{k|a\le |k|\le b\}$, 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 $\sigma >2$.

24 pages