Electron. J. Diff. Eqns., Vol. 2008(2008), No. 143, pp. 1-18.

Generalized eigenfunctions of relativistic Schrodinger operators in two dimensions

Tomio Umeda, Dabi Wei

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.

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Tomio Umeda
Department of Mathematical Science, University of Hyogo
Shosha 2167, Himeji 671-2201, Japan
email: umeda@sci.u-hyogo.ac.jp
Dabi Wei
Department of Mechanical and Control Engineering
Graduate School of Science and Engineering
Tokyo Institute of Technology
2-12-1 S5-22 O-okayama, Meguro-ku, Tokyo 152-8550, Japan
email: dabi@ok.ctrl.titech.ac.jp

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