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

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

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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 languageEnglish
Pages (from-to)1587-1623
Number of pages37
JournalCommunications in Partial Differential Equations
Issue number9
StatePublished - 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].


  • Distributional potentials
  • Inverse scattering
  • Miura potentials
  • Schrödinger operators

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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