## 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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