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.