Recovery of singularities for formally determined inverse problems
This paper deals with the Schrödinger equation in \(\mathbb{R}^ 2\) for potentials of compact support in \(L^ \infty\), and with the acoustic equation in \(\mathbb{R}^ 2\) with a bounded obstacle in which the sound speed is in \(L^ \infty\). The authors prove that if the scattering amplitude for the Schrödinger equation is given for all angles and one energy, the potential is determined to within a function in \(C^ \alpha(\mathbb{R}^ 2)\), \(\forall\alpha\), \(0\leq \alpha<1\), where \(C^ \alpha (\mathbb{R}^ 2)\) is the Hölder space of order \(\alpha\). Furthermore, the potential (again to within the same possible ambiguity) is equal to the inverse Fourier transform of the function \(T_ q\) used by Beals and Coifman and by Ablowitz and Nachman in the \(\overline{\partial}\) approach. They then rely on the fact that the Dirichlet-to-Neumann map determines \(T_ q\) to suggest that the location and strengths of the singularities of \(q\) can be obtained from that map by the method of Nachman. Similarly, the authors prove that if the values of the fundamental solutions of the acoustic equation on the surface of the obstacle are given, the sound speed is determined to within a function in \(C^ \alpha(\mathbb{R}^ 2)\), \(\forall\alpha\), \(0\leq\alpha<1\).
scientific article; zbMATH DE number 205355
scientific article