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\({\mathcal{PT}}\) -Symmetric Waveguides and the Lack of Variational Techniques

  • David Krejčiřík
Open Problems

Abstract

On a model of waveguide with non-Hermitian Robin-type boundary conditions, we demonstrate the need for a robust method establishing the existence of isolated eigenvalues for non-self-adjoint operators possessing both essential and discrete spectrum.

Mathematics Subject Classification (2010)

35P15 35J05 47B44 81Q12 

Keywords

Robin Laplacian non-self-adjoint boundary conditions complex symmetric operator \({\mathcal{PT}}\) -symmetry waveguides discrete and essential spectra 

References

  1. 1.
    Borisov D., Krejčiřík D.: \({\mathcal{PT} }\) -symmetric waveguides. Integral Equ. Oper. Theory 62, 489–515 (2008)zbMATHCrossRefGoogle Scholar
  2. 2.
    Krejčiřík D., Kříž J.: On the spectrum of curved quantum waveguides. Publ. RIMS, Kyoto University. 41, 757–791 (2005)zbMATHCrossRefGoogle Scholar
  3. 3.
    Krejčiřík D., Tater M.: Non-Hermitian spectral effects in a \({\mathcal{PT}}\) -symmetric waveguide. J. Phys. A Math. Theor. 41, 244013 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Theoretical Physics, Nuclear Physics InstituteAcademy of SciencesŘežCzech Republic

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