The following two claims are proved: If \(n\) is a compactly supported function (with support \(B\)), \(\text{Im }n>0\) in \(B\), \(k=\text{const}>0\), and some regularity conditions hold, then
1) the problem \((\nabla^ 2+ k^ 2)v=0\) in \(B\), \((\nabla^ 2+ n)w=0\) in \(B\), \(u= w\) and \(u_ N= w_ N\) on \(\partial B\), \(N\) is the outer normal to \(\partial B\), has only the trivial solution; and
2) the set \(\{u-w, u_ N- w_ N\}\) is dense in \(L^ 2(\partial B)\times L^ 2(\partial B)\) when \(u=\int_{S^ 2} g(\alpha)\exp(- ik\alpha\cdot x)d\alpha\), \(g(\alpha)\) runs through all of \(L^ 2(S^ 2)\) and \(w\) runs through all of \(\{w: (\nabla^ 2+ n)w=0\) in \(B\), \(w\in H^ 2(B)\}\).
Remarks: 1) On p. 30, line 1, a reference is made to Corollary 3.9 of the second author’s dissertation to justify a conclusion which is a standard consequence of the elliptic estimates and embedding theorems.
2) The authors suggest a variational problem (VP) for solving the inverse scattering problem (ISP) similar to the one in the paper by the first author and P. Monk [Q. J. Mech. Appl. Math. 41, No. 1, 97-125 (1988; Zbl 0637.73026)], and other papers of these authors. One of the drawbacks of their approach is the lack of equivalence between ISP and VP. In [the reviewer, Appl. Math. Lett. 7, No. 2, 57-61 (1994)] a VP equivalent to ISP is given.