Towards an Inverse Scattering theory for non decaying potentials of the heat equation

The resolvent approach is applied to the spectral analysis of the heat equation with non decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe $N$ solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a DBAR-problem explicitly in terms of the corresponding objects associated to the original potential. Regularity conditions of the potential in the cases N=1 and N=2 are investigated in details. The singularities of the resolvent for the case N=1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.