Spectral properties of the Schrödinger operator with operator-valued potential

Let H be a separable Hilbert space, and let H := L₂(R, H). In the space H, we consider the self-adjoint Schrödinger operator of the formTq f = -f'' + qf,where q is a reflectionless operator-valued potential.Let P₊ and P₋ be the spectral projectors of the operator Tq corresponding to the positive half-line R₊ and the negative half-line R₋, respectively. Define H_± := P_± H, and let Tq^± := Tq|H_±. In this paper, we show that the operator Tq has trivial kernel, and that the operator Tq⁺ is unitarily equivalent to the unperturbed operator T₀. Next, let B be an arbitrary bounded negative operator in a separable Hilbert space H₁ (H₁ ≤ H if H < ∞). Then we prove that there exists a reflectionless potential q such that Tq⁻ is unitarily equivalent to the operator B. A key role in this work is played by solutions of the operator Riccati equation of the formS'(x) = K S(x) + S(x) K - 2 S(x) K S(x), x ∈ R,where K ∈ B₊(H) \0\, and S R → B(H). Here, B(H) is the Banach algebra of all bounded linear operators acting in H, and B₊(H) = ∈ B(H) A ≥ 0\.