Inverse scattering on the line for Schrödinger operators with Miura potentials, II. Different Riccati representatives

2009/10/04 by R. O. Hryniv, Hryniv, R. O., Ya. V. Mykytyuk +3
Mathematics · Physics and Astronomy · #34L25 (Primary) #34L40 #47L10 #81U40 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:34L25 #msc:34L40 #msc:47L10 #msc:81U40

paper · pdf · doi:10.48550/arxiv.0910.0639

32 pages

arxiv created 2009/10/04 · arxiv updated 2009/12/01

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∈ 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.

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