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

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.