## 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 language | English |
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Title of host publication | Operator Methods in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2010 |

Editors | Jan Janas, Pavel Kurasov, Ari Laptev, Sergei Naboko |

Pages | 1-42 |

Number of pages | 42 |

DOIs | |

State | Published - 2013 |

Event | 5th International Conference: Operator Theory, Analysis and Mathematical Physics, OTAMP 2010 - Bedlewo, Poland Duration: Aug 5 2010 → Aug 12 2010 |

### Publication series

Name | Operator Theory: Advances and Applications |
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Volume | 227 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 2296-4878 |

### Conference

Conference | 5th International Conference: Operator Theory, Analysis and Mathematical Physics, OTAMP 2010 |
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Country/Territory | Poland |

City | Bedlewo |

Period | 8/5/10 → 8/12/10 |

### Bibliographical note

Publisher Copyright:© 2013 Springer Basel.

## Keywords

- Impedance function
- Inverse scattering problem
- Schrodinger operator

## ASJC Scopus subject areas

- Analysis