# Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions - Mathematics > Spectral Theory

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Abstract: Generalized eigenfunctions of the two-dimensional relativistic Schr\-odingeroperator $H=\sqrt{-\Delta}+Vx$ with $|Vx|\leq C< x>^{-\sigma}$,$\sigma>3-2$, are considered. We compute the integral kernels of the boundaryvalues $R 0^\pm\lambda=\sqrt{-\Delta}-\lambda\pm i0^{-1}$, and prove thatthe generalized eigenfunctions $\phi^\pmx,k$ are bounded on $R x^2\times\{k |a\leq |k|\leq b\}$, where $a,b\subset0,\infty\backslash\sigma pH$, and$\sigma pH$ is the set of eigenvalues of $H$. With this fact and thecompleteness of the wave operators, we establish the eigenfunction expansionfor the absolutely continuous subspace for $H$. Finally, we show that eachgeneralized eigenfunction is asymptotically equal to a sum of a plane wave anda spherical wave under the assumption that $\sigma>2$.

Author: ** Tomio Umeda University of Hyogo, Dabi Wei Tokyo Institute of Technology**

Source: https://arxiv.org/