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


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.


Inverse Problem Riemann Surface Negative Energy Scattering Problem Real Potential 


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© Plenum Publishing Corporation 1994

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  • P. G. Grinevich

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