The problem of recovering a potential q(y) in the differential equation:
−∆u + q(y)u = 0 (x,y) &∈ (0, 1) × (0,1)
u(0, y) = u(1, y) = u(x, 0) = 0
u(x, 1) = f(x), uy(x, 1) = g(x)
is investigated. The method of separation of variables reduces the recovery 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 characterize the ill-posedness of this inverse problem in terms of the "distinguishability" of potentials. An estimate is derived which indicates the maximum level of measurement error under which two potentials, differing only far away from y = 1, can be resolved.
Copyright © 1994 OPA (Overseas Publishers Association). This article first appeared in Applicable Analysis 55, no. 3-4 (1994): 157-75. doi:10.1080/00036819408840296.
Please note that downloads of the article are for private/personal use only.
Caudill, Lester F. "Determination of a Potential from Cauchy Data: Uniqueness and Distinguishability." Applicable Analysis 55, no. 3-4 (1994): 157-75. doi:10.1080/00036819408840296.