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

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.