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 TMF, 1994, Volume 99, Number 2, Pages 300–308 (Mi tmf1590)

Fast decaying potentials on the finite-gap background and the $\bar \partial$-problem on the Riemann surfaces

P. G. Grinevich

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: The direct and the inverse ‘scattering problems’ for the heat-conductivity operator $L_P=\partial_y-\partial_x^2+u(x,y)$ are studied for the following class of potentials: $u(x,y)=u_0(x,y)+u_1(x,y)$ where $u_0(x,y)$ is a nonsingular real finite-gap potential and $u_1(x,y)$ decays sufficiently fast as $x^2+y^2 \rightarrow \infty$. We show that the ‘scattering data’ for such potentials is the $\bar \partial$-problem data on the Riemann surface corresponding to the potential $u_0(x,y)$. The ‘scattering data’ corresponding to real potentials is characterized and it is proved that the inverse problem corresponding to such data has unique nonsingular solution without the ‘small norm’ assumption. Analogs of these results for the fixed negative energy scattering problem for the two-dimensional time-independent Schrödinger operator $L_P=-\partial _x^2-\partial _y^2+u(x,y)$ are obtained.

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Theoretical and Mathematical Physics, 1994, 99:2, 599–605

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Citation: P. G. Grinevich, “Fast decaying potentials on the finite-gap background and the $\bar \partial$-problem on the Riemann surfaces”, TMF, 99:2 (1994), 300–308; Theoret. and Math. Phys., 99:2 (1994), 599–605

Citation in format AMSBIB
\Bibitem{Gri94} \by P.~G.~Grinevich \paper Fast decaying potentials on the finite-gap background and the $\bar \partial$-problem on the Riemann surfaces \jour TMF \yr 1994 \vol 99 \issue 2 \pages 300--308 \mathnet{http://mi.mathnet.ru/tmf1590} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1308792} \zmath{https://zbmath.org/?q=an:0850.35081} \transl \jour Theoret. and Math. Phys. \yr 1994 \vol 99 \issue 2 \pages 599--605 \crossref{https://doi.org/10.1007/BF01016145} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PV07100017} 

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This publication is cited in the following articles:
1. P. G. Grinevich, “Scattering transformation at fixed non-zero energy for the two-dimensional Schrödinger operator with potential decaying at infinity”, Russian Math. Surveys, 55:6 (2000), 1015–1083
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