Complex Analysis and Operator Theory

, Volume 8, Issue 1, pp 255–281 | Cite as

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

  • David Krejčiřík
  • Petr Siegl
  • Jakub Železný


We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.


Sturm–Liouville operators Non-symmetric Robin boundary conditions Similarity to normal or self-adjoint operators Discrete spectral operator Complex symmetric operator \(\mathcal{PT }\)-symmetry Metric operator \(\mathcal{C }\) operator Hilbert–Schmidt operators 

Mathematics Subject Classification (2010)

Primary 34B24 47B40 34L10 Secondary 34L40 34L05 81Q12 



D.K. acknowledges the hospitality of the Deusto Public Library in Bilbao. This work has been partially supported by the Czech Ministry of Education, Youth, and Sports within the project LC06002 and by the GACR grant No. P203/11/0701. P.S. appreciates the support by GACR grant No. 202/08/H072 and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS OHK4-010/10. J.Ž. appreciates the support by the Czech Ministry of Education, Youth, and Sports within the project LC527.


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • David Krejčiřík
    • 1
    • 2
  • Petr Siegl
    • 1
    • 3
    • 4
  • Jakub Železný
    • 1
    • 3
  1. 1.Department of Theoretical Physics, Nuclear Physics InstituteCzech Academy of Sciences ŘežCzech Republic
  2. 2.IKERBASQUE, Basque Foundation for ScienceBilbaoSpain
  3. 3.Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePragueCzech Republic
  4. 4.Laboratoire Astroparticule et CosmologieParisFrance

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