An inverse problem for Helmholtz's equation. III (Q749781)

[For part II see Inverse Probl. 3, L 59-L 61 (1987; Zbl 0654.35088).] Consider the \(problem\) [\(\Delta\) \(+k^ 2n(z)]u=-\delta (x)h(k)\), \(x=(x_ 1,x_ 2,x_ 3)\), \(z=x_ 3\), \(h(k)=\int^{T}_{0}a(t)\exp (ikt)dt\), Im a\(=0\), \(a\in L^ 1[0,T]\), \(u'_+=\alpha u'_ -\), \(u_+=u_ -\) on \(P:=\{x|\) \(z=0\}\), \(u'_{\pm}:=(\partial u/\partial z)|_{z=\pm 0}\), \(\alpha =const>0.\) Given the surface data u(x,k) at \(z=0\) for all \(x\in P\) and all \(k>0\), one seeks to recover h(k), n(z) and \(\alpha\). The function n(z) is known for \(z>0\) and \(n(z)\in L^{\infty}(-\infty,0)\) is unknown. It is proved that the surface data determine uniquely and analytically \(\{\) n(z),h(k),\(\alpha\}\) if one of the following pieces of information is given: i) the total energy \(E=\int^{T}_{0}a^ 2dt\) of the wavelet, or ii) the value n(-0), or iii) the value n(-\(\infty).\) In formula (24) one term is missing so that the text below (24) should be modified.