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Register a new userInverse scattering on the line for Schrodinger operators with Miura potentials, II. Different Riccati representatives
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Added by
Rostyslav O. Hryniv
Publication date
2009
fields
Physics
and research's language is
English
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This is the second in a series of papers on scattering theory for one-dimensional Schrodinger operators with Miura potentials admitting a Riccati representation of the form $q=u+u^2$ for some $uin L^2(R)$. 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 Schrodinger 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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This is the first in a series of papers on scattering theory for one-dimensional Schrodinger operators with highly singular potentials $qin H^{-1}(R)$. In this paper, we study Miura potentials $q$ associated to positive Schrodinger operators that admit a Riccati representation $q=u+u^2$ for a unique $uin L^1(R)cap L^2(R)$. 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 $S:umapsto r$ is real-analytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.
We study the direct and inverse scattering problem for the one-dimensional Schrodinger equation with steplike potentials. We give necessary and sufficient conditions for the scattering data to correspond to a potential with prescribed smoothness and prescribed decay to their asymptotics. These results are important for solving the Korteweg-de Vries equation via the inverse scattering transform.
We consider the Schrodinger operator $H_{eta W} = -Delta + eta W$, self-adjoint in $L^2({mathbb R}^d)$, $d geq 1$. Here $eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{eta W}$ near the origin, and due to the irregular decay of $eta W$, we encounter some non semiclassical phenomena. In particular, $H_{eta W}$ has less eigenvalues than suggested by the semiclassical intuition.
We show that a generic quasi-periodic Schrodinger operator in $L^2(mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrodinger operator with the resulting potential has empty absolutely continuous spectrum.
We consider the Schrodinger operator on $[0,1]$ with potential in $L^1$. We prove that two potentials already known on $[a,1]$ ($ain(0,{1/2}]$) and having their difference in $L^p$ are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of $p$ (and $a$) showing the role of $p$.
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