Spectral properties of the Schrödinger operator with operator-valued potential - Mykytyuk, Ya. V.; Sushchyk, N. S. (Matematychni Studii, 2025)
Let H be a separable Hilbert space, and let H := L_2(R, H). In the space H, we consider the self-adjoint Schrödinger operator of the form T_q f = -f’’ + qf, where q is a reflectionless operator-valued potential. Let P_+ and P_- be the spectral projectors of the operator T_q corresponding to the positive half-line R_+ and the negative half-line R_-, respectively. Define H_± := P_± H, and let T_q^± := T_q|_H_±. In this paper, we show that the operator T_q has trivial kernel, and that the operator T_q^+ is unitarily equivalent to the unperturbed operator T_0. Next, let B be an arbitrary bounded negative operator in a separable Hilbert space H_1 (H_1 H if H < ∞). Then we prove that there exists a reflectionless potential q such that T_q^- is unitarily equivalent to the operator B. A key role in this work is played by solutions of the operator Riccati equation of the form S’(x) = K S(x) + S(x) K - 2 S(x) K S(x), x R, where K B_+(H) {0}, and SR → B(H). Here, B(H) is the Banach algebra of all bounded linear operators acting in H, and B_+(H) = {A B(H) A ≥ 0}.