Impossibility of recovering a scatterer’s shape by the first version of the “linear sampling” method

The version of the “linear sampling” method introduced in D. Colton, A. Kirsch, Inverse Problems, 12, 383, 1996, is analyzed. It consists into the attempt to recover a scatterer's contour by determining an indicator function from the knowledge of the scattered far field data. In particular, the indicator function is determined following the solution of a certain far field equation. We show that the far field equation to solve does not have (almost anywhere) a solution for two classes of objects: Perfectly conducting cylinders and homogeneous dielectric cylinders having circular cross section. It thus follows that the indicator function is “infinite” (almost) everywhere both inside and outside the scatterer and, consequently, does not represent the shape of the object.