Tomio Umeda, Dabi Wei
Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions
(234K, pdf)
ABSTRACT. Generalized eigenfunctions of the two-dimensional
relativistic Schr\"o\-dinger operator
$H=\sqrt{-\Delta}+V(x)$ with $|V(x)|\leq C\langle x\rangle^{-\sigma}$,
$\sigma>3/2$, are considered.
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 $\sigma>2$.