Heat operator with pure soliton potential: properties of Jost and dual Jost solutions

TL;DR

This paper investigates the analytical and asymptotic properties of Jost solutions for the heat equation with pure solitonic potentials, revealing boundedness and annihilator regions in the spectral parameter space.

Contribution

It provides a detailed analysis of Jost solutions' properties and identifies regions where the extended heat operator has left and right annihilators.

Findings

Boundedness of certain transformed Jost solutions in polygonal regions of the q-plane.

Identification of regions where the extended heat operator has annihilators.

Detailed asymptotic behavior of solutions on the x-plane.

Abstract

Properties of Jost and dual Jost solutions of the heat equation, $\Phi(x,k)$ and $\Psi(x,k)$, in the case of a pure solitonic potential are studied in detail. We describe their analytical properties on the spectral parameter $k$ and their asymptotic behavior on the $x$-plane and we show that the values of $e^{-qx}\Phi(x,k)$ and the residua of $e^{qx}\Psi(x,k)$ at special discrete values of $k$ are bounded functions of $x$ in a polygonal region of the $q$-plane. Correspondingly, we deduce that the extended version $L(q)$ of the heat operator with a pure solitonic potential has left and right annihilators for $q$ belonging to these polygonal regions.