RECOVERING TWO COEFFICIENTS IN AN ELLIPTIC EQUATION VIA PHASELESS INFORMATION

For fixed y is an element of R-3, we consider the equation Lu + k(2)u = -delta(x - y), x is an element of R-3, where L = div(n(x)(-2)del) q(x), k > 0 is a frequency, n(x) is a refraction index and q(x) is a potential. Assuming that the refraction index n(x) is different from 1 only inside a bounded compact domain Omega with a smooth boundary S and the potential q(x) vanishes outside of the same domain, we study an inverse problem of finding both coefficients inside Omega from some given information on solutions of the elliptic equation. Namely, it is supposed that the point source located at point y is an element of S is a variable parameter of the problem. Then for the solution u(x, y, k) of the above equation satisfying the radiation condition, we assume to be given the following phaseless information f(x, y, k) = |u(x, y, k)|(2) for all x, y is an element of S and for all k >= k(0) > 0, where ko is some constant. We prove that this phaseless information uniquely determines both coefficients n(x) and q(x) inside Omega.