Stability of the inverse problem for the Helmholtz equation (Q1920770)

This paper studies an inverse problem for the Helmholtz equation \[ \Delta u+ \lambda^2 a(x)u= \delta(x-x_0) \] in \(\mathbb{R}^3\), where \(a(x)=1 +b(x)\) and \(b(x)\) is continuous with compact support. The particular problem is to determine \(b(x)\), given some information about the scattered field. The paper establishes a stability theorem under certain auxiliary conditions on \(b(x)\). The proof is based on the reduction of the multidimensional problem to a family of one-dimensional problems and uses techniques of analytic continuation for a function of a single variable.