A LOCAL BORG-MARCHENKO THEOREM FOR COMPLEX POTENTIALS


R WEIKARD

rudi@math.uab.edu

Department of Mathematics
University of Alabama at Birmingham
University Station
Birmingham AL 35294, USA






Consider the Sturm-Liouville problem given by the equation $ -y''+qy=\lambda y$ on $ [0,\infty)$ and the boundary condition $ y(0)=0$. The famous Borg-Marchenko theorem states that the associated Weyl-Titchmarsh $ m$-function determines uniquely the potential $ q$ when $ q$ is real. The local version states that $ q$ is determined on $ [0,a]$ if the $ m$-function is known up to errors of the order of $ \exp(-2a\Re(\sqrt{-\lambda}))$ (where $ \Re(\sqrt{-\lambda})$ is positive) as $ \lambda$ tends to infinity along some non-real ray.

We show that under certain very general restrictions on $ q$ an analogous result holds also for complex-valued potentials.

This is joint work with B. M. Brown and R. A. Peacock.


EPSRC Gregynog Workshop, 21-26 July 2002

Gregynog Abstracts