Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 127, pp. 1-46.
Title: Generalized eigenfunctions of relativistic
Schrodinger operators I
Author: Tomio Umeda (Univ. of Hyogo, Japan)
Abstract:
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 $(\sqrt{-\Delta}-\lambda \mp i0)^{-1}$,
$\lambda >0$, 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 $\sigma >3$.
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.