Spectral properties of the Schrödinger operator with operator-valued potential

Анотація

Let $H$ be a separable Hilbert space, and let $\mathcal{H} := L_2(\mathbb{R}, H)$. In the space $\mathcal{H}$, we consider the self-adjoint Schrödinger operator of the form
$
T_q f = -f'' + qf,
$
where $q$ is a reflectionless operator-valued potential.
Let $\mathcal{P}_+$ and $\mathcal{P}_-$ be the spectral projectors of the operator $T_q$ corresponding to the positive half-line $\mathbb{R}_+$ and the negative half-line $\mathbb{R}_-$, respectively. Define $\mathcal{H}_\pm := \mathcal{P}_\pm \mathcal{H}$, and let $T_q^\pm := T_q|_{\mathcal{H}_\pm}$.

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 $H_1$ ($\dim H_1 \le \dim H$ if $\dim H < \infty$). 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
$S'(x) = K S(x) + S(x) K - 2 S(x) K S(x), \qquad x \in \mathbb{R},$
where $K \in \mathcal{B}_+(H) \setminus \{0\}$, and $S\colon \mathbb{R} \to \mathcal{B}(H)$. Here, $\mathcal{B}(H)$ is the Banach algebra of all bounded linear operators acting in $H$, and $\mathcal{B}_+(H) = \{A \in \mathcal{B}(H) \mid A \geq 0\}$.