On spectral theory for Schrödinger operators with operator-valued potentials

Academic Article

Abstract

  • Given a complex, separable Hilbert space H, we consider differential expressions of the type τ=-(d2/dx2)+V(x), with x∈(a, ∞) or x∈R. Here V denotes a bounded operator-valued potential V({dot operator})∈B(H) such that V({dot operator}) is weakly measurable and the operator norm {norm of matrix}V({dot operator}){norm of matrix}B(H) is locally integrable.We consider self-adjoint half-line L2-realizations Hα in 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)=0, indexed by the self-adjoint operator α=α*∈B(H). In addition, we study self-adjoint full-line L2-realizations H of τ in L2(R;dx;H). In either case we treat in detail basic spectral theory associated with Hα and H, including Weyl-Titchmarsh theory, Green's function structure, eigenfunction expansions, diagonalization, and a version of the spectral theorem. © 2013 Elsevier Inc.
  • Authors

    Published In

    Digital Object Identifier (doi)

    Author List

  • Gesztesy F; Weikard R; Zinchenko M
  • Start Page

  • 1784
  • End Page

  • 1827
  • Volume

  • 255
  • Issue

  • 7