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Journal d'Analyse Mathématique

, Volume 137, Issue 1, pp 373–427 | Cite as

Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

  • Fritz GesztesyEmail author
  • Sergey N. Naboko
  • Rudi Weikard
  • Maxim Zinchenko
Article
  • 22 Downloads

Abstract

Given a complex, separable Hilbert space \(\mathcal{H}\), we consider differential expressions of the type τ = −(d2/dx2)\(I_\mathcal{H}\) + V(x), with x ∈ (x0,∞) for some x0 ∈ ℝ, or x ∈ ℝ (assuming the limit-point property of τ at ±∞). Here V denotes a bounded operator-valued potential V(·) ∈ \(\mathcal{B}(\mathcal{H})\) such that V(·) is weakly measurable, the operator norm \(||V(\cdot)||_{\mathcal{B}(\mathcal{H})}\) is locally integrable, and V(x) = V(x)* a.e. on x ∈ [x0,∞) or x ∈ ℝ. We focus on two major cases. First, on m-function theory for self-adjoint half-line L2-realizations H+,α in L2((x0,∞); dx;\(\mathcal{H}\)) (with x0 a regular endpoint for τ, associated with the self-adjoint boundary condition sin(α)u′(x0) + cos(α)u(x0) = 0, indexed by the selfadjoint operator α = α* ∈ \(\mathcal{B}(\mathcal{H})\)), and second, on m-function theory for self-adjoint full-line L2-realizations H of τ in L2(ℝ; dx;\(\mathcal{H}\)).

In a nutshell, a Donoghue-type m-function \(M^{Do}_{A,\mathcal{N}{_i}}(\cdot)\) associated with self-adjoint extensions A of a closed, symmetric operator \(\dot{A}\) in \(\mathcal{H}\) with deficiency spaces Nz = ker (\(\dot{A}\) * −zI\(\mathcal{H}\)) and corresponding orthogonal projections \({P_{{N_z}}}\) onto Nz is given by
$$\begin{gathered} M_{A,{\mathcal{N}_i}}^{Do}(z) = {P_{{\mathcal{N}_i}}}(zA + {I_\mathcal{H}})(A) - z{I_\mathcal{H}}{)^{ - 1}}{P_{{\mathcal{N}_i}|{\mathcal{N}_i}}} \hfill \\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = z{I_{{\mathcal{N}_i}}} + ({z^2} + 1){P_{{N_i}}}{(A - z{I_\mathcal{H}})^{ - 1}}{P_{{N_i}|{\mathcal{N}_i}}},\;\;\;z \in \mathbb{C}\backslash \mathbb{R}. \hfill \\ \end{gathered}$$

In the concrete case of half-line and full-line Schrödinger operators, the role of \(\dot{A}\) is played by a suitably defined minimal Schrödinger operator H+,min in L2((x0,∞); dx;\(\mathcal{H}\)) and Hmin in L2(ℝ; dx;\(\mathcal{H}\)), both of which will be proven to be completely non-self-adjoint. The latter property is used to prove that if H+,α in L2((x0,∞); dx;\(\mathcal{H}\)), respectively, H in L2(ℝ; dx;\(\mathcal{H}\)), are self-adjoint extensions of H+,min, respectively, Hmin, then the corresponding operator-valued measures in the Herglotz–Nevanlinna representations of the Donoghue-type m-functions \(M^{Do}_{H+,\alpha},\mathcal{N_{+,i}}(\cdot)\) and \(M^{Do}_{H,\mathcal{N}{_i}}(\cdot)\) encode the entire spectral information of H+,α, respectively, H.

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Fritz Gesztesy
    • 1
    Email author
  • Sergey N. Naboko
    • 2
  • Rudi Weikard
    • 3
  • Maxim Zinchenko
    • 4
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of Mathematical PhysicsSt. Petersburg State UniversitySt. PetersburgRussian Federation
  3. 3.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA
  4. 4.Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerqueUSA

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