ON A CLASS OF INTEGRAL-EQUATIONS OF THE 1ST KIND IN INVERSE SCATTERING-THEORY

The authors recently introduced a constructive method for solving the inverse obstacle problem and the inverse inhomogeneous medium problem for acoustic waves [D. Colton and P. Monk, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 278-291], [D. Colton and P. Monk, Quart. J. Mech. Appl. Math., 41 (1988), pp. 97-125]. If F(theta; alpha) is the far field pattern corresponding to the incident field exp [ikr cos (theta - alpha)] with wave number k > 0, their method is based on the fact that the integral operator K:L2[-pi, pi] --> L2[-pi, pi] defined by Kg = integral-pi/-pi F(theta; alpha)g(theta) dtheta is injective and has dense range. Unfortunately, there can exist values of k such that this is not the case for either of the inverse problems considered in the above-mentioned references. Motivated by an idea of Douglas Jones for the direct obstacle problem for acoustic waves [D. Jones, Quart. J. Mech. Appl. Math., 27 (1974), pp. 129-142], it is shown how to modify the operator K such that the modified operator is injective with dense range for all values of k > 0. The proof of this shows how the method of Colton and Monk can be modified to be applicable for all positive values of the wave number k.