Inverse scattering for non-classical impedance Schrödinger operators

Sergio Albeverio, Rostyslav O. Hryniv, Yaroslav V. Mykytyuk, Peter A. Perry

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 (Formula presented) 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.

Original languageEnglish
Title of host publicationOperator Methods in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2010
EditorsJan Janas, Pavel Kurasov, Ari Laptev, Sergei Naboko
Pages1-42
Number of pages42
DOIs
StatePublished - 2013
Event5th International Conference: Operator Theory, Analysis and Mathematical Physics, OTAMP 2010 - Bedlewo, Poland
Duration: Aug 5 2010Aug 12 2010

Publication series

NameOperator Theory: Advances and Applications
Volume227
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Conference

Conference5th International Conference: Operator Theory, Analysis and Mathematical Physics, OTAMP 2010
Country/TerritoryPoland
CityBedlewo
Period8/5/108/12/10

Bibliographical note

Publisher Copyright:
© 2013 Springer Basel.

Keywords

  • Impedance function
  • Inverse scattering problem
  • Schrodinger operator

ASJC Scopus subject areas

  • Analysis

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