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

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

2009-10-04

arxiv.org/abs/0910.0639v1

Mathematics

Spectral Theory

32 pages

Scientific paper

This is the second in a series of papers on scattering theory for one-dimensional Schr\"odinger operators with Miura potentials admitting a Riccati representation of the form $q=u'+u^2$ 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 Schr\"odinger 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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