Jost solutions of Schrodinger operators with reflectionless operator- valued potentials - Mykytyuk, Ya.; Sushchyk, N. (Matematychni Studii, 2025)
Let H be a separable Hilbert space, and let H be the Hilbert space of square integrable functions fR→ H. In this paper, we consider the reflectionless Schr"odinger operator T_qf=-f’’+qf acting in H and study the corresponding Jost solutions, i.e., solutions of the equation -y’’+qy=λ^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), xR, where KB_+(H){0}, SR→ B(H). Here B(H) is the Banach algebra of all linear continuous operators acting in H, and B_+(H)={AB(H)A≥ 0}.