Building an extended resolvent of the heat operator via twisting transformations

We introduce twisting transformations for the heat operator. By the simultaneous use of these transformations, N solitons are superimposed à la Darboux on a generic smooth potential decaying at infinity, and the corresponding Jost solutions are generated. We also use these twisting operators to study the existence of the related extended resolvent. We study the existence and uniqueness of the extended resolvent in detail in the case of N solitons with N "incoming" rays and one "outgoing" ray.