03-454 Tomio Umeda

Generalized eigenfunctions of relativistic Schr "odinger operators I (174K, LaTeX 2e) Oct 6, 03

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Abstract. Generalized eigenfunctions of the 3-dimensional relativistic Schr "o dinger 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$

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