  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.
