ON THE CONSTRUCTION OF A POTENTIAL FROM CAUCHY DATA

We investigate the problem of recovering a potential q(y) in the differential equation: -DELTAu+q(y)u = 0, (x,y) is-an-element-of (0,1)X(0,1), u(0,y) = u(1,y) = u(x,0) = 0, u(x,1) = f(x), u(y)(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.