96-373 F. Gesztesy and B. Simon

Inverse Spectral Analysis with Partial Information on the Potential, I. The Case of an A.C. Component in the Spectrum (20K, amstex) Aug 19, 96

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Abstract. We consider operators $-\frac{d^2}{dx^2} + V$ in $L^2 (\Bbb R)$ with the sole hypothesis that $V$ is limit point at $\pm\infty$ and that $-\frac{d^2}{dx^2} + V$ in $L^2 ((0,\infty))$ has some absolutely continuous component $S_+$ in its spectrum. We prove that $V$ on $(-\infty,0)$ is completely determined by knowledge of $V$ on $(0,\infty)$ and by the reflection coefficient $R_+(\lambda)$ for scattering from right incidence and energies $\lambda \in S$, where $S \subseteq S_+$ has positive Lebesgue measure.

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