Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type

Abstract.

The spectral structure of two parameter unbounded operator pencils of waveguide type is studied. Theorems on discreteness of the spectrum for a fixed parameter are proved. Variational principles for real eigenvalues in some parts of the root zones are established. In the case of n = 1 (quadratic pencils) domains containing the spectrum are described (see Fig. 1–3). Conditions in the definition of the pencils of waveguide type arise naturally from physical problems and each of them has a physical meaning. In particular a connection between the energetic stability condition and a perturbation problem for the coefficients is given.

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Correspondence to N. Çolakoğlu.

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Çolakoğlu, N., Hasanov, M. & Uzun, B.Ü. Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type. Integr. equ. oper. theory 56, 381–400 (2006). https://doi.org/10.1007/s00020-006-1429-1

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Mathematics Subject Classification (2000).

  • Primary 47A75
  • Secondary 49R50
  • 34L15

Keywords.

  • Waveguide
  • operator pencil
  • spectral sets and eigenvalues
  • variational principles