Determination of a potential from cauchy data: uniqueness and distinguishability

Abstract

The problem of recovering a potential q(y) in the differential equation:

is investigated. The method of separation of variables reduces the recovcry of q(y) to a non–standard inverse Sturm–Liouville problem. Employing

asymptotic techniques and integral operators of Gel'fand–Levitan type, it is shown that, under appropriate conditions on the Cauchy pair (f,g), q(y) is uniquely determined, in a local sense, up to its mean. We charactercrize the ill–posedness of this inverse problem in terms of the “distinguishability” of poetentials. An estimate is derived which indicates the maximum level of measurement error under which two potentials, dffering only far away from y=1, can be resolved.

Keywords:

• inverse problem
• undetermined coefficient
• Overposed boundary data