Inverse scattering for schrödinger operators with miura potentials, II. different riccati representatives

Rostyslav O. Hryniv; Yaroslav V. Mykytyuk; Peter A. Perry

doi:10.1080/03605302.2011.593014

Inverse scattering for schrödinger operators with miura potentials, II. different riccati representatives

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

Mathematics

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u' + u² for some u Ie{cyrillic, ukrainian} L²(R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L₁(R (1 +⌊x⌋)dx) generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.

Original language	English
Pages (from-to)	1587-1623
Number of pages	37
Journal	Communications in Partial Differential Equations
Volume	36
Issue number	9
DOIs	https://doi.org/10.1080/03605302.2011.593014
State	Published - Sep 2011

Bibliographical note

Funding Information:
This material is based upon work supported by the National Science Foundation under Grant DMS-0710477 (RH and PP) and by the Deutsche Forschungsge-meinschaft under project 436 UKR 113/84 (RH and YM). RH acknowledges support from the College of Arts and Sciences at the University of Kentucky and thanks the Department of Mathematics at the University of Kentucky for its hospitality during his stay there. RH, YM, and PP thank the Institut für Angewandte Mathematik der Universität Bonn for hospitality during part of the time that this work was done. PP thanks Percy Deift for helpful conversations and SFB 611 for financial support of his research visit to Universität Bonn. The authors thank Iryna Egorova for bringing to their attention the paper [11].

Keywords

Distributional potentials
Inverse scattering
Miura potentials
Schrödinger operators

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/03605302.2011.593014

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abstract = "This is the second in a series of papers on scattering theory for one-dimensional Schr{\"o}dinger operators with Miura potentials admitting a Riccati representation of the form q = u' + u2 for some u Ie{cyrillic, ukrainian} L2(R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L1(R (1 +⌊x⌋)dx) generating positive Schr{\"o}dinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.",

keywords = "Distributional potentials, Inverse scattering, Miura potentials, Schr{\"o}dinger operators",

author = "Hryniv, {Rostyslav O.} and Mykytyuk, {Yaroslav V.} and Perry, {Peter A.}",

note = "Funding Information: This material is based upon work supported by the National Science Foundation under Grant DMS-0710477 (RH and PP) and by the Deutsche Forschungsge-meinschaft under project 436 UKR 113/84 (RH and YM). RH acknowledges support from the College of Arts and Sciences at the University of Kentucky and thanks the Department of Mathematics at the University of Kentucky for its hospitality during his stay there. RH, YM, and PP thank the Institut f{\"u}r Angewandte Mathematik der Universit{\"a}t Bonn for hospitality during part of the time that this work was done. PP thanks Percy Deift for helpful conversations and SFB 611 for financial support of his research visit to Universit{\"a}t Bonn. The authors thank Iryna Egorova for bringing to their attention the paper [11].",

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AU - Hryniv, Rostyslav O.

AU - Mykytyuk, Yaroslav V.

AU - Perry, Peter A.

N1 - Funding Information: This material is based upon work supported by the National Science Foundation under Grant DMS-0710477 (RH and PP) and by the Deutsche Forschungsge-meinschaft under project 436 UKR 113/84 (RH and YM). RH acknowledges support from the College of Arts and Sciences at the University of Kentucky and thanks the Department of Mathematics at the University of Kentucky for its hospitality during his stay there. RH, YM, and PP thank the Institut für Angewandte Mathematik der Universität Bonn for hospitality during part of the time that this work was done. PP thanks Percy Deift for helpful conversations and SFB 611 for financial support of his research visit to Universität Bonn. The authors thank Iryna Egorova for bringing to their attention the paper [11].

PY - 2011/9

Y1 - 2011/9

N2 - This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u' + u2 for some u Ie{cyrillic, ukrainian} L2(R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L1(R (1 +⌊x⌋)dx) generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.

AB - This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u' + u2 for some u Ie{cyrillic, ukrainian} L2(R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L1(R (1 +⌊x⌋)dx) generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.

KW - Distributional potentials

KW - Inverse scattering

KW - Miura potentials

KW - Schrödinger operators

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Inverse scattering for schrödinger operators with miura potentials, II. different riccati representatives

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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