Electron. J. Diff. Eqns., Vol. 2006(2006), No. 127, pp. 1-46.

Generalized eigenfunctions of relativistic Schrodinger operators I

Tomio Umeda

Generalized eigenfunctions of the 3-dimensional relativistic Schrodinger operator $\sqrt{-\Delta} + V(x)$ with $|V(x)|\le C \langle x \rangle^{{-\sigma}}$, $\sigma > 1$, are considered. We construct the generalized eigenfunctions by exploiting results on the limiting absorption principle. We compute explicitly the integral kernel of $(\sqrt{-\Delta}-z)^{-1}$, $z \in {\mathbb C}\setminus [0, +\infty)$, which has nothing in common with the integral kernel of $({-\Delta}-z)^{-1}$, but the leading term of the integral kernels of the boundary values $(\sqrt{-\Delta}-\lambda \mp i0)^{-1}$, $\lambda$ greater than 0, turn out to be the same, up to a constant, as the integral kernels of the boundary values $({-\Delta}-\lambda \mp i0)^{-1}$. This fact enables us to show that the asymptotic behavior, as $|x| \to +\infty$, of the generalized eigenfunction of $\sqrt{-\Delta} + V(x)$ is equal to the sum of a plane wave and a spherical wave when $\sigma$ greater than 3.

Submitted September 22, 2006. Published October 11, 2006.
Math Subject Classifications: 35P99, 35S99, 47G30, 47A40.
Key Words: Relativistic Schrodinger operators; pseudo-relativistic Hamiltonians; generalized eigenfunctions; Riesz potentials; radiation conditions.

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Tomio Umeda
Department of Mathematical Sciences
University of Hyogo
Shosha, Himeji 671-2201, Japan
Telephone +81-792-67-4935 Fax +81-792-66-8868
email: umeda@sci.u-hyogo.ac.jp

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