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

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 $q=u^{\prime }+u^2$ for some $u\in L^2(R)$. 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 $L^1(R,(1+|x|)dx)$ 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 $r$.

32 pages

References in corpus (1)

• Inverse spectral problems for Dirac operators with summable potentials

Cited by in corpus (1)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials