 11112 Odile Bastille, Alexei Rybkin
 On the determinant formula in the inverse scattering procedure with a partially known steplike potential
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Aug 10, 11

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Abstract. We are concerned with the inverse scattering problem for the full line Schr dinger operator ∂_{x} +q(x) with a steplike potential q a priori known on R₊=(0,∞). Assuming q_{R₊} is known and short range, we show that the unknown part q_{R₋} of q can be recovered by
q_{R₋}(x)=2∂_{x} logdet(1+(1+M_{x}⁺)⁻ G_{x}),
where M_{x}⁺ is the classical Marchenko operator associated to q_{R₊} and G_{x} is a trace class integral Hankel operator. The kernel of G_{x} is explicitly constructed in term of the difference of two suitably defined reflection coefficients. Since q_{R₋} is not assumed to have any pattern of behavior at ∞, defining and analyzing scattering quantities becomes a serious issue. Our analysis is based upon some subtle properties of the TitchmarshWeyl mfunction associated with R₋.
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