Inverse Scattering for Non-classical Impedance Schrödinger Operators

Abstract

We review recent progress in the direct and inverse scattering theory for one-dimensional Schrödinger operators in impedance form. Two classes of non-smooth impedance functions are considered. Absolutely continuous impedances correspond to singular Miura potentials that are distributions from $W^{-1}_{2,loc}(\mathbb{R})$; nevertheless, most of the classic scattering theory for Schrödinger operators with Faddeev–Marchenko potentials is carried over to this singular setting, with some weak decay assumptions. The second class consists of discontinuous impedances and generates Schrödinger operators with unusual scattering properties. In the model case of piece-wise constant impedance functions with discontinuities on a periodic lattice the corresponding reflection coefficients are periodic. In both cases, a complete description of the scattering data is given and the explicit reconstruction method is derived.

Mathematics Subject Classification (2010). Primary: 34L25, Secondary: 34L40,47L10, 81U40.