Extended resolvent and inverse scattering with an application to KPI

We present in detail an extended resolvent approach for investigating linear problems associated to 2 + 1 dimensional integrable equations. Our presentation is based as an example on the nonstationary Schrödinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of the inverse scattering theory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given.