Property C for ODE and Applications to an InverseProblem for. a Heat Equation.

摘要

Let = —d2 /dx2 + k2q,(x), k = const > 0, j = 1, 2, 0 < ess inf q_j(x) < ess sup q_j (x) < oo. Suppose that (*) 0 p(x)ut (x, k)u2 (x, k) dx = 0 for all k > 0, where p is an arbitrary fixed bounded piecewise-analytic function on [0, 1], which changes sign finitely many times, and u, solves the problem ,u_j = 0, 0 < x < 1, 14 (0, k) = 0, u, (0, k) = 1. It is proved that (*) implies p = 0. This result is applied to an inverse problem for a heat equation.