Skip to main content
SpringerLink
Log in

The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

The non-equivalence of the presence of antilinear symmetry and pseudo-Hermiticity is shown for bounded operators. Two appropriate examples are operators with non-empty residual spectrum. The class of operators for which the equivalence holds is extended to the spectral operators of scalar type. The importance of J-self-adjointness is stressed and new proofs using this property are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ali Mostafazadeh, J. Math. Phys. 43, 3944 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. G Scolarici and L Solombrino, On the pseudo-Hermitian nondiagonalizable Hamiltonians, quant-ph/0211161 (2002)

  3. L Solombrino, Weak pseudo-Hermiticity and antilinear commutant, quantph/0203101 (2002)

  4. D E Edmunds and W D Evans, Spectral theory and differential operators, Oxford Mathematical Monographs (Oxford University Press, USA, 1987)

    MATH  Google Scholar 

  5. Denis Borisov and David Krejčiřík, Integral Equations Operator Theory 62(4), 489 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Stephan Ramon Garcia, The eigenstructure of complex symmetric operators, Proceedings of the Sixteenth International Conference on Operator Theory and Applications (IWOTA 16) Vol. 179, pp. 169–183 (2008)

    ADS  Google Scholar 

  7. Stephan Ramon Garcia and Mihai Putinar, Trans. Am. Math. Soc. 358, 1285 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Stephan Ramon Garcia and Mihai Putinar, Trans. Am. Math. Soc. 359, 3913 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Emil Prodan, Stephan Ramon Garcia and Mihai Putinar, J. Phys. A: Math. Gen. 39(2), 389 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Petr Siegl, Quasi-Hermitian models, Master’s thesis (Faculty of Nuclear Sciences and Physical Engineering, CTU, Prague) ssmf.fjfi.cvut.cz/2008/siegl thesis.pdf, 2007/2008

  11. Jiří Blank, Pavel Exner and Miloslav Havlíček, Hilbert Space Operators in Quantum Physics, Computational and Applied Mathematical Physics (American Institute of Physics, 1999)

  12. Ali Mostafazadeh, J. Math. Phys. 47, 092101 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  13. Michael Reed and Barry Simon, Methods of modern mathematical physics, in: Functional analysis (Academic Press, revised and enlarged edition, 1980) Vol. 1

  14. Nelson Dunford and Jacob T Schwartz, Linear operators-Part III Spectral Operators (Wiley-Interscience, 1971)

  15. Ali Mostafazadeh, J. Math. Phys. 43, 2814 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. David Krejčiřík, Hynek Bíla and Miloslav Znojil, J. Phys. A: Math. Gen. 39, 10143 (2006)

    Article  MATH  ADS  Google Scholar 

  17. Jean Dieudonné, Quasi-Hermitian operators, Proceedings of the International Symposium on Linear Spaces, July 1961, pp. 115–123

Download references

Author information

Authors and Affiliations

  1. Faculty of Nuclear Sciences and Physical Engineering, Prague, Czech Republic

    Petr Siegl

  2. Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, Řež, Czech Republic

    Petr Siegl

  3. Laboratoire Astroparticule et Cosmologie, Université Paris 7, Paris, France

    Petr Siegl

Authors
  1. Petr Siegl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr Siegl.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Siegl, P. The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry. Pramana - J Phys 73, 279–286 (2009). https://doi.org/10.1007/s12043-009-0119-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12043-009-0119-3

Keywords

PACS Nos

Search

Navigation