An operator Riccati equation and reflectionless Schrodinger operators

Abstract

In this paper, we study a connection between the operator Riccati equation

$\displaystyle
S'(x)=KS(x)+S(x)K-2S(x)KS(x), \quad x\in\mathbb{R},
$

and the set of reflectionless Schr\"odinger operators with operator-valued potentials.
Here $K\in \mathcal{B}(H)$, $K>0$ and $S:\mathbb{R}\to \mathcal{B}(H)$, where $\mathcal{B}(H)$ is the Banach algebra of all linear continuous operators acting in a separable Hilbert space $H$. Let $\mathscr{S}^+(K)$ be the set of all solutions $S$ of the Riccati equation satisfying the conditions $0< S(0)< I $ and $S'(0)\ge 0$, with $I$ being the identity operator in $H$. We show that every solution $S\in \mathscr{S}^+(K)$ generates a reflectionless Schr\"odinger operator with some potential $q$ that is an analytic function in the strip

$\displaystyle
\Pi_K:=\left\{z=x+iy \mid x,y\in\mathbb{R}, \,\, |y|<\tfrac{\pi}{2\|K\|} \right\};
$

moreover,

$\displaystyle \|q(x+iy)\|\le2\|K\|^2\cos^{-2}(y\|K\|), \quad (x+iy)\in\Pi_K .
$