On spectral theory for Schrödinger operators with operator-valued potentials

Given a complex, separable Hilbert space HH, we consider differential expressions of the type τ=−(d2/dx2)+V(x)τ=−(d2/dx2)+V(x), with x∈(a,∞)x∈(a,∞) or x∈Rx∈R. Here V   denotes a bounded operator-valued potential V(⋅)∈B(H)V(⋅)∈B(H) such that V(⋅)V(⋅) is weakly measurable and the operator norm ‖V(⋅)‖B(H)‖V(⋅)‖B(H) is locally integrable.We consider self-adjoint half-line L2L2-realizations HαHα in L2((a,∞);dx;H)L2((a,∞);dx;H) associated with τ, assuming a   to be a regular endpoint necessitating a boundary condition of the type sin(α)u′(a)+cos(α)u(a)=0sin(α)u′(a)+cos(α)u(a)=0, indexed by the self-adjoint operator α=α⁎∈B(H)α=α⁎∈B(H). In addition, we study self-adjoint full-line L2L2-realizations H of τ   in L2(R;dx;H)L2(R;dx;H). In either case we treat in detail basic spectral theory associated with HαHα and H, including Weyl–Titchmarsh theory, Greenʼs function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem.