On a Class of Integral Equations of the First Kind in Inverse Scattering Theory

Abstract

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. App!. 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 ${\bf K}:L^2 [ - \pi ,\pi ] \to L^2 [ - \pi ,\pi ]$ defined by ${\bf K}g = \int _{ - \pi }^\pi F( {\theta ;\alpha } )g( \theta )d\theta $ 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 ${\bf K}$ 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.