Abstract

A Uniqueness Theorem in Inverse Spectral Theory C. Bennewitz Really satisfactory results in inverse spectral theory have so far only been obtained for the simplest Sturm-Liouville equation $-u''+qu=\lambda u$ and some closely related equations. A more general equation $-(pu')'+qu=\lambda wu$ may sometimes be transformed to this form (`Liouville transformation'), but this requires considerable smoothness of the coefficients. One would therefore like to deal with the more general equation directly. An immediate difficulty is then that given such an equation there are always many unitarily equivalent equations of the same form, again via Liouville transformations. Inverse spectral theory should therefore attempt to reconstruct the equivalence class of the equation from spectral data, two equations being equivalent if they can be transformed into each other unitarily via a Liouville transformation. This seems to be a difficult problem to attack in full generality, and a first step might be to show that the potential $q$ is uniquely determined by appropriate spectral data, provided the coefficients $p$ and $w$ are known. I will report on such a result, which requires little smoothness of the coefficients and also allows arbitrary sign changes in $p$. The existing proofs for the case $p=w=1$ rest heavily on the fact that the solutions of the corresponding equation for large $\lambda$ are asymptotic to exponentials, the special properties of which are then crucial to the arguments. We will therefore use a different approach, where much weaker estimates on the logarithmic derivatives of the solutions, valid for $\lambda$ on strictly non-real rays, are used. These estimates are based on earlier results of the author.

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