R. O. Hryniv
On norm resolvent convergence of Schr\"odinger operators with $\delta'$-like potentials.
Fri, 11/06/2009 - 16:01 — SomNambulAuthors: R. O. Hryniv, Yu. D. Golovaty
Subjects: Spectral Theory
AbstractWe address the problem on the right definition of the Schroedinger operator
with potential $\delta'$, where $\delta$ is the Dirac delta-function. Namely,
we prove the uniform resolvent convergence of a family of Schroedinger
operators with regularized short-range potentials $\epsilon^{-2}V(x/\epsilon)$
tending to $\delta'$ in the distributional sense as $\epsilon\to 0$. In 1986,
P.Inverse scattering on the line for Schr\"odinger operators with Miura potentials, II. Different Riccati representatives.
Tue, 10/06/2009 - 17:01 — SomNambulAuthors: R. O. Hryniv, Ya. V. Mykytyuk, P. A. Perry
Subjects: Spectral Theory
AbstractThis 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.Inverse scattering on the line for Schr\"odinger operators with Miura potentials, II. Different Riccati representatives.
Tue, 10/06/2009 - 17:01 — SomNambulAuthors: R. O. Hryniv, Ya. V. Mykytyuk, P. A. Perry
Subjects: Spectral Theory
AbstractThis 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.Inverse scattering for Schr\"odinger operators with Miura potentials, I. Unique Riccati representatives and ZS-AKNS systems.
Tue, 10/06/2009 - 17:01 — SomNambulAuthors: C. Frayer, R. O. Hryniv, Ya. V. Mykytyuk, P. A. Perry
Subjects: Spectral Theory
AbstractThis is the first in a series of papers on scattering theory for
one-dimensional Schr\"odinger operators with highly singular potentials $q\in
H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive
Schr\"odinger operators that admit a Riccati representation $q=u'+u^2$ for a
unique $u\in L^1(R)\cap L^2(R)$. Such potentials have a well-defined reflection
coefficient $r(k)$ that satisfies $|r(k)|<1$ and determines $u$ uniquely. We
show that the scattering map $S:u\mapsto r$ is real-analytic with real-analytic
inverse.
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