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

Properties of Jost and dual Jost solutions of the heat equation, Φ(x, k) and Ψ(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−qxΦ(x, k) and the residues of eqxΨ(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.