Building extended resolvent of heat operator via twisting transformations

TL;DR

This paper introduces twisting transformations for the heat operator to superimpose solitons, generate Jost solutions, and analyze the existence and uniqueness of the extended resolvent in soliton scenarios.

Contribution

It develops a method using twisting transformations to construct solitons and study the extended resolvent of the heat operator with detailed analysis for specific soliton configurations.

Findings

Existence of the extended resolvent is established for N solitons.

Uniqueness of the extended resolvent is proved under certain conditions.

The method provides explicit construction of Jost solutions for soliton potentials.

Abstract

Twisting transformations for the heat operator are introduced. They are used, at the same time, to superimpose a` la Darboux N solitons to a generic smooth, decaying at infinity, potential and to generate the corresponding Jost solutions. These twisting operators are also used to study the existence of the related extended resolvent. Existence and uniqueness of the extended resolvent in the case of $N$ solitons with N "ingoing" rays and one "outgoing" ray is studied in details.