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 Bäcklund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a ∂̄-problem explicitly in terms of the corresponding objects associated with the original potential. Regularity conditions of the potential in the cases N = 1 and 2 are investigated in detail. 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.