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

Research paper by **Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko**

Indexed on: **07 Sep '11**Published on: **07 Sep '11**Published in: **Mathematics - Spectral Theory**

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#### 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.