Fast-decaying potentials on the finite-gap background and thebar partial - problem on the Riemann surfaces
The direct and the inverse ‘scattering problems’ for the heat-conductivity operatorLP = partial _y - partial _x^2 + u(x,y) are studied for the following class of potentials: u(x,y)=u o (x,y)+u 1( x,y), where u o ( x,y) is a nonsingular real finite-gap potential and u 1( x,y) decays sufficiently fast as x 2+y2→∞. We show that the ‘scattering data’ for such potentials is thebar partial - data on the Riemann surface corresponding to the potential u 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 operatorLP = - partial _x^2 - partial _y^2 + u(x,y) are obtained.