 0069 Michael Christ, Alexander Kiselev
 WKB Asymptotics of Generalized Eigenfunctions
of OneDimensional Schr\"odinger Operators
(51K, LaTeX)
Feb 10, 00

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Abstract. We prove the WKB asymptotic
behavior of solutions of the differential equation
$d^2u/dx^2+V(x)u=k^2u$ in two cases. First, for a.e.\ $k^2$ when
$V \in L^p(\reals)$, where $1 \leq p <2$. Second, for a.e.\ $k^2>A$
when $V \in L^{\infty}(\reals)$ and $V' \in L^p(\reals)$,
$1 \leq p <2$, where $A = \limsup_{x \rightarrow \infty}V(x)$.
These results imply that Schr\"odinger operators
with such potentials have absolutely continuous spectrum
on $(0, \infty)$ ($(A, \infty)$ in the second case).
We also establish WKB asymptotic behavior
of solutions for some energydependent potentials.
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