# On Spectral Theory for Schr\"odinger Operators with Operator-Valued
Potentials

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

Indexed on: **15 Mar '13**Published on: **15 Mar '13**Published in: **Mathematics - Spectral Theory**

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#### Abstract

Given a complex, separable Hilbert space $\cH$, we consider differential
expressions of the type $\tau = - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$
or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot)
\in \cB(\cH)$ such that $V(\cdot)$ is weakly measurable and the operator norm
$\|V(\cdot)\|_{\cB(\cH)}$ is locally integrable.
We consider self-adjoint half-line $L^2$-realizations $H_{\alpha}$ in
$L^2((a,\infty); dx; \cH)$ associated with $\tau$, assuming $a$ to be a regular
endpoint necessitating a boundary condition of the type $\sin(\alpha)u'(a) +
\cos(\alpha)u(a)=0$, indexed by the self-adjoint operator $\alpha = \alpha^*
\in \cB(\cH)$. In addition, we study self-adjoint full-line $L^2$-realizations
$H$ of $\tau$ in $L^2(\bbR; dx; \cH)$. In either case we treat in detail basic
spectral theory associated with $H_{\alpha}$ and $H$, including
Weyl--Titchmarsh theory, Green's function structure, eigenfunction expansions,
diagonalization, and a version of the spectral theorem.