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Hamiltonian Systems with Eigenvalue Depending Boundary Conditions

  • Aad Dijksma
  • Heinz Langer
  • Henk de Snoo
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 35)

Abstract

In earlier papers [DLS1–6] we have described the selfadjoint extensions, in indefinite inner product spaces and with nonempty resolvent sets of a symmetric closed relation S in a Hilbert space ℌ by means of generalized resolvents, characteristic functions and Štraus extensions. In this paper we show how these results can be applied when S comes from a 2n×2n Hamiltonian system of ordinary differential equations on an interval [a, b), (1.1) Jy′(t) = (ℓΔ(t) + H(t)) y(t) + Δ(t)f(t), t ∈[a, b), ℓ∈ℂ, which is regular in a and in the limit point case in b; for further specifications see Section 5. We pay special attention to selfadjoint extensions beyond the given space ℌ, as they give rise to eigenvalue and boundary value problems with boundary conditions of the form (1.2) A(ℓ)y 1 (a) + B(ℓ)y 2 (a) = 0, in which the matrix coefficients A(ℓ) and В(ℓ) depend holomorphically on the eigenvalue parameter , see Theorem 7.1 below. The eigenvalue problem for such a selfadjoint extension in a larger space can be considered as a linearization of the corresponding boundary value problem (1.1) and (1.2).

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Copyright information

© Birkhäuser Verlag Basel 1988

Authors and Affiliations

  • Aad Dijksma
    • 1
  • Heinz Langer
    • 2
  • Henk de Snoo
    • 1
  1. 1.Wiskunde En InformaticaRijksuniversiteit GroningenGroningenNederland
  2. 2.Sektion MathematikTechnische UniversitätDresdenGermany

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