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Networks and Heterogeneous Media

2011, Volume 6, Issue 2: 257-277. doi: 10.3934/nhm.2011.6.257
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Spectral theory for nonconservative transmission line networks

  • 1.
    Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80933
  • Received: 01 August 2010 Revised: 01 April 2011

  • Primary: 34B45.

  • The global theory of transmission line networks with nonconservative junction conditions is developed from a spectral theoretic viewpoint. The rather general junction conditions lead to spectral problems for nonnormal operators. The theory of analytic functions which are almost periodic in a strip is used to establish the existence of an infinite sequence of eigenvalues and the completeness of generalized eigenfunctions. Simple eigenvalues are generic. The asymptotic behavior of an eigenvalue counting function is determined. Specialized results are developed for rational graphs.

    Citation: Robert Carlson. Spectral theory for nonconservative transmission line networks[J]. Networks and Heterogeneous Media, 2011, 6(2): 257-277. doi: 10.3934/nhm.2011.6.257

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  • The global theory of transmission line networks with nonconservative junction conditions is developed from a spectral theoretic viewpoint. The rather general junction conditions lead to spectral problems for nonnormal operators. The theory of analytic functions which are almost periodic in a strip is used to establish the existence of an infinite sequence of eigenvalues and the completeness of generalized eigenfunctions. Simple eigenvalues are generic. The asymptotic behavior of an eigenvalue counting function is determined. Specialized results are developed for rational graphs.


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