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Direct and inverse scattering problem for one-dimensional perturbed Hill operator. (Russian) Zbl 0616.34017

The author considers the direct and inverse scattering problem for the following pair of operators: \(H_ 0=-d^ 2/dx^ 2+p(x),\) \(H=H_ 0+q(x)\), \(Im p(x)=Im q(x)=0,\) \(p(x+1)=p(x),\) \(x\in {\mathbb{R}}\), \(p\in L_ 2(0,1)\), \(Q_ r=\int_{R}| q(x)| (1+| x|^ r)dx<\infty,\) \(r=1\) or \(r=2\). The conditions which are necessary for \(Q_ 2<\infty\) and sufficient for the existence of a unique potential with given scattering characteristics and finite first moment \(Q_ 1<\infty\) are given.
Reviewer: D.Herceg

MSC:

34L99 Ordinary differential operators
35P25 Scattering theory for PDEs
34D10 Perturbations of ordinary differential equations
47A55 Perturbation theory of linear operators
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