The authors study uniqueness of recovery of a bounded simply connected domain with from its scattering amplitude corresponding to the Helmholtz equation outside a soft on or hard on impenetrable obstacle or to the equation in describing a penetrable obstacle .
They correct the Schiffer’s proof of uniqueness for soft and give a first uniqueness proof for hard by using the reviewer’s idea of exploiting singular solutions suggested in the paper [V. Isakov, Commun. Pure Appl. Math. 41, No. 7, 865-877 (1988; Zbl 0676.35082)]. Also they give a simpler proof of uniqueness of a penetrable scatterer than in the reviewer’s previous paper [Commun. Partial Differ. Equations 15, No. 11, 1565-1587 (1990; Zbl 0728.35148)].
For other uniqueness results in the inverse scattering we refer to the book [D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (1992; Zbl 0760.35053)], and to the review paper [V. Isakov, Uniqueness and stability in multidimensional inverse problems, Inverse Problems 9, 579-621 (1993)].