On the Construction of a Potential from Cauchy Data

323 Caudill Jr, L.F. and B.D. Lowe, On the construction of a potential from Cauchy data, Journal of Computational and Applied Mathematics 47 (1993) 323-333. We investigate the problem of recovering a potential q(y) in the differential equation: Llu + q(y)u = 0, (x, y)E(O, l)x(O, 1), u(O, y)=u(l, y)=u(x,0)=0, u(x, l)= f(x), u/x, 1)= g(x). The method of separation of variables reduces the recovery of q( y) to a nonstandard inverse Sturm-Liouville problem. An asymptotic formula is developed that suggests that under appropriate conditions on the Cauchy pair (f, g ), q( y) is uniquely determined up to the mean. Moreover, the recovery of q( y) is comparable to finding a function from its polynomial moments. A reconstruction scheme is suggested and numerical examples are considered.