11-112 Odile Bastille, Alexei Rybkin
On the determinant formula in the inverse scattering procedure with a partially known steplike potential (63K, LaTeX 2e) 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 Titchmarsh-Weyl m-function associated with R₋.

Files: 11-112.src( 11-112.keywords , BastilleRybkin11detsteplike.tex )