Initial Value Problems and Weyl--Titchmarsh Theory for Schr\"odinger Operators with Operator-Valued Potentials

Abstract

We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a complex, separable Hilbert space. We assume regularity of the left endpoint $a$ and the limit point case at the right endpoint $b$. In addition, the bounded self-adjoint operator $\alpha= \alpha^* \in \cB(\cH)$ is used to parametrize the self-adjoint boundary condition at the left endpoint $a$ of the type $$ \sin(\alpha)u'(a)+\cos(\alpha)u(a)=0, $$ with $u$ lying in the domain of the underlying maximal operator $H_{\max}$ in $L^2((a,b);dx;\cH)$ associated with $\tau$. More precisely, we establish the existence of the Weyl-Titchmarsh solution of $H_{\alpha}$, the corresponding Weyl-Titchmarsh $m$-function $m_{\alpha}$ and its Herglotz property, and determine the structure of the Green's function of $H_{\alpha}$. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient $V$, -y" + (V - z) y = f \, \text{on} \, (a,b), y(x_0) = h_0, \; y'(x_0) = h_1, under the following general assumptions: $(a,b)\subseteq\bbR$ is a finite or infinite interval, $x_0\in(a,b)$, $z\in\bbC$, $V:(a,b)\to\cB(\cH)$ is a weakly measurable operator-valued function with $\|V(\cdot)\|_{\cB(\cH)}\in L^1_\loc((a,b);dx)$, and $f\in L^1_{\loc}((a,b);dx;\cH)$, with $\cH$ a complex, separable Hilbert space. We also study the analog of this initial value problem with $y$ and $f$ replaced by operator-valued functions $Y, F \in \cB(\cH)$. Our hypotheses on the local behavior of $V$ appear to be the most general ones to date.