On Spectral Theory for Schrödinger Operators with Operator-Valued Potentials

Abstract

Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $\tau =-(d^2/d{x}^2)+V(x)$, with $x\in (a,\infty )$ or $x\in \mathbb{R}$. 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 $\sin (\alpha )u^{\prime }(a)+\cos (\alpha )u(a)=0$, indexed by the self-adjoint operator $α= α^* \in \cB(\cH)$. 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.

49 pages. arXiv admin note: substantial text overlap with arXiv:1109.1613, arXiv:1111.0645, arXiv:math/0505120