Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

Academic Article

Abstract

  • We develop Weyl-Titchmarsh theory for self-adjoint Schrodinger operators Hα in L2((a,b);dx;H) associated with the operator-valued differential expression τ = - (d2/dx2) + V(·), with V: (a, b) → B(H), and H a complex, separable Hilbert space. We assume regularity of the left endpoint a and the limit point case at the right endpoint b. In addition, the bounded self-adjoint operator α = α* ∈ B(H) is used to parametrize the self-adjoint boundary condition at the left endpoint a of the type sin(α)u'(a) + cos(α)u(a) = 0, with u lying in the domain of the underlying maximal operator Hmax in L2((a,b);dx;H) associated with τ. More precisely, we establish the existence of the Weyl-Titchmarsh solution of Hα, the corresponding Weyl-Titchmarsh m -function mα and its Herglotz property, and determine the structure of the Green's function of Hα. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient V, under the following general assumptions: (a,b) ⊆ R is a finite or infinite interval, x0 6 (a,b), z ∈ C, V: (a,b) → B(H) is a weakly measurable operator-valued function with ||V(·)||B(H) ∈ L1loc((a,b);dx), and f ∈ L1loc((a,b);dx;H). We also study the analog of this initial value problem with y and f replaced by operator-valued functions Y, F ∈ B(H). Our hypotheses on the local behavior of V appear to be the most general ones to date.
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    Author List

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

  • 241
  • End Page

  • 283
  • Volume

  • 7
  • Issue

  • 2