Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 143, pp. 1-18.
Title: Generalized eigenfunctions of relativistic Schrodinger operators
in two dimensions
Authors: Tomio Umeda (Univ. of Hyogo, Himeji, Japan)
Dabi Wei (Tokyo Institute of Technology, Tokyo, Japan)
Abstract:
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.