An operator Riccati equation and reflectionless Schrodinger operators

In this paper, we study a connection between the operator Riccati equation S'(x)=KS(x)+S(x)K-2S(x)KS(x), x, and the set of reflectionless Schr\"odinger operators with operator-valued potentials.Here K∈ B(H), $K>0$ and S:R→ B(H), where B(H) is the Banach algebra of all linear continuous operators acting in a separable Hilbert space H. Let S⁺(K) be the set of all solutions S of the Riccati equation satisfying the conditions $0< S(0)< I $ and S'(0)≥ 0, with I being the identity operator in H. We show that every solution S∈ S⁺(K) generates a reflectionless Schr\"odinger operator with some potential q that is an analytic function in the strip ΠK:==x+iy x,y, \,\, |y|<π/2\|K\| \; moreover, \|q(x+iy)\|≤2\|K\|²cos⁻²(y\|K\|), (x+iy)∈ΠK .