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

Abstract:
This article concerns the generalized eigenfunctions of the two-dimensional relativistic Schrodinger operator with , . We compute the integral kernels of the boundary values , and prove that the generalized eigenfunctions are bounded on , where , and is the set of eigenvalues of . With this fact and the completeness of the wave operators, we establish the eigenfunction expansion for the absolutely continuous subspace for . 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 .

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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