2017, Behrndt, Jussi, Malamud, Mark M., Neidhardt, Hagen

A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.

2016, Bao, Gang, Hu, Guanghui, Yin, Tao, Sun, Jiguang

Assume a time-harmonic elastic wave is incident onto a penetrable anisotropic body embedded into a homogeneous isotropic background medium. We propose an equivalent variational formulation in a truncated bounded domain and show uniqueness and existence of weak solutions by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the scattering interface is derived. As an application, we design a descent algorithm for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are demonstrated to show the validity and accuracy of our methods.