 08153 Tomio Umeda, Dabi Wei
 Generalized eigenfunctions of relativistic Schroedinger operators in two dimensions
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Aug 26, 08

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Abstract. Generalized eigenfunctions of the twodimensional
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 $\sigma>2$.
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