Jost solutions of Schrodinger operators with reflectionless operator- valued potentials

Abstract

Let $H$ be a separable Hilbert space, and let $\mathcal{H}$ be the Hilbert space of square integrable functions $f\colon \mathbb{R}\to H$. In this paper, we consider the reflectionless Schr\"odinger operator $T_qf=-f''+qf$ acting in $\mathcal{H}$ and study the corresponding Jost solutions, i.e., solutions of the equation
$$-y''+qy=\lambda^2 y$$
with a reflectionless operator-valued potential $q$. In particular, we provide an explicit formula for the Jost solutions in terms of solutions of the Riccati equation $$S'(x)=KS(x)+S(x)K-2S(x)KS(x), \qquad x\in\mathbb{R},$$
where $K\in \mathcal{B}_+(H)\setminus\{0\}$, $S\colon \mathbb{R}\to \mathcal{B}(H)$. Here $\mathcal{B}(H)$ is the Banach algebra of all linear continuous operators acting in $H$, and $\mathcal{B}_+(H)=\{A\in \mathcal{B}(H)\mid A\geq 0\}$.