Inverse scattering theory of the heat equation for a perturbed one-soliton potential

The inverse scattering theory of the heat equation is developed for a special subclass of potentials nondecaying at space infinity - perturbations of the one-soliton potential by means of decaying two-dimensional functions. Extended resolvent, Green's functions, and Jost solutions are introduced and their properties are investigated in detail. The singularity structure of the spectral data is given and then the inverse problem is formulated in an exact distributional sense.