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Theoretical and Mathematical Physics

, Volume 99, Issue 2, pp 599–605 | Cite as

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

  • P. G. Grinevich
Article
  • 31 Downloads

Abstract

The direct and the inverse ‘scattering problems’ for the heat-conductivity operator\(LP = \partial _y - \partial _x^2 + u(x,y)\) are studied for the following class of potentials:u(x,y)=u o (x,y)+u1(x,y), whereu o (x,y) is a nonsingular real finite-gap potential andu1(x,y) decays sufficiently fast asx2+y2→∞. We show that the ‘scattering data’ for such potentials is the\(\bar \partial - \) data on the Riemann surface corresponding to the potentialu o (x,y). The ‘scattering data’ corresponding to real potentials is characterized and it is proved that the inverse problem corresponding to such data has a 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\(LP = - \partial _x^2 - \partial _y^2 + u(x,y)\) are obtained.

Keywords

Inverse Problem Riemann Surface Negative Energy Scattering Problem Real Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • P. G. Grinevich

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