  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.
