Integral Equations and Operator Theory

, Volume 77, Issue 3, pp 303–354 | Cite as

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

  • Sergio Albeverio
  • Mark Malamud
  • Vadim MogilevskiiEmail author


We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)−B(t)y(t) = Δ(t) f(t) on an interval \({\mathcal{I}=[a,b) }\) with the regular endpoint a. It is assumed that the deficiency indices n ±(T min) of the minimal relation T min associated with this system in \({L^2_\Delta(\mathcal{I})}\) satisfy \({n_-(T_{\rm min})\leq n_+(T_{\rm min})}\). We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size \({N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}\). Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform \({V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}\) where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension \({\tilde{T}}\) of T min induced by the boundary problem is unitarily equivalent to the multiplication operator in L 2(Σ). Hence the multiplicity of spectrum of \({\tilde{T}}\) does not exceed N Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.

Mathematics Subject Classification (2010)

34B08 34B20 34B40 34L10 47A06 47B25 


First-order symmetric system nevanlinna boundaryconditions m-function spectral function of a boundary problem fourier transform 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Sergio Albeverio
    • 1
    • 2
  • Mark Malamud
    • 3
  • Vadim Mogilevskii
    • 4
    Email author
  1. 1.Institut für Angevandte Mathematic HCM, IZKS, SFB611Universität BonnBonnGermany
  2. 2.CERFIMLocarnoSwitzerland
  3. 3.Institute of Applied Mathematics and MechanicsNAS of UkraineDonetskUkraine
  4. 4.Department of Mathematical AnalysisLugans’k National UniversityLugans’kUkraine

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