Fastdecaying potentials on the finitegap background and thebar partial  problem on the Riemann surfaces
Abstract
The direct and the inverse ‘scattering problems’ for the heatconductivity 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 finitegap potential and u _{1}( x,y) decays sufficiently fast as x ^{2}+y^{2}→∞. 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 twodimensional timeindependent Schrödinger operatorLP =  partial _x^2  partial _y^2 + u(x,y) are obtained.
 Publication:

Theoretical and Mathematical Physics
 Pub Date:
 May 1994
 DOI:
 10.1007/BF01016145
 Bibcode:
 1994TMP....99..599G