[go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

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

  • Published:
View saved research
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

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.

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

Access this article

Log in via an institution

Subscribe and save

Springer+
from €37.37 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

Notes

  1. \(A\) is positive if \(\langle f, A f \rangle > 0\) for all \(f \in \mathcal{H }, f\ne 0\).

References

  1. Adduci, J., Mityagin, B.: Eigensystem of an \(L^2\)-perturbed harmonic oscillator is an unconditional basis. Central Eur. J. Math. 10, 569–589 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Albeverio, S., Kuzhel, S.: One-dimensional Schrödinger operators with \({\cal{P}}\)-symmetric zero-range potentials. J. Phys. A Math. Gen. 38, 4975–4988 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Albeverio, S., Motovilov, A., Shkalikov, A.: Bounds on variation of spectral subspaces under \(J\)-self-adjoint perturbations. Integral Equ. Oper. Theory 64, 455–486 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benamara, N.-E., Nikolski, N.: Resolvent tests for similarity to a normal operator. Proc. Lond. Math. Soc. 78, 585–626 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)

    Article  Google Scholar 

  6. Bender, C.M., Boettcher, S., Meisinger, P.: PT-symmetric quantum mechanics. J. Math. Phys. 40, 2201–2229 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)

    Article  MathSciNet  Google Scholar 

  8. Borisov, D., Krejčiřík, D.: PT-symmetric waveguides. Integral Equ. Oper. Theory 62, 489–515 (2008)

    Google Scholar 

  9. Ćurgus, B., Najman, B.: The operator \((\text{sgn } x)\frac{\text{ d }^2}{\text{ d }x^2}\) is similar to a selfadjoint operator in \(L^2({\mathbb{R}})\). Proc. Am. Math. Soc. 123, 1125–1128 (1995)

    MATH  Google Scholar 

  10. Davies, E.B.: Spectral theory and differential operators. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  11. Dieudonné, J.: Quasi-Hermitian operators. In: Proceedings of the International Symposium on Linear Spaces, pp. 115–123 (1961)

  12. Djakov, P., Mityagin, B.: Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators J. Funct. Anal. 263, 2300–2332 (2012)

    Google Scholar 

  13. Dunford, N., Schwartz, J.T.: Linear operators, Part 3. In: Spectral Operators. Wiley, New York (1971)

  14. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators (Oxford Mathematical Monographs). Oxford University Press, USA (1987)

    Google Scholar 

  15. Faddeev, M., Shterenberg, R.: On the similarity of some differential operators to self-adjoint ones. Math. Notes 72, 261–270 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gesztesy, F., Tkachenko, V.: A criterion for Hill operators to be spectral operators of scalar type. J. Anal. Math. 107, 287–353 (2009)

    Google Scholar 

  17. Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space. AMS, Providence (1969)

    Google Scholar 

  18. Günther, U., Kuzhel, S.: \({\cal{PT}}\)-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras. J. Phys. A Math. Theoret. 43, 392002 (2010)

    Article  Google Scholar 

  19. Hernandez-Coronado, H., Krejčiřík, D., Siegl, P.: Perfect transmission scattering as a \({\cal{PT}}\)-symmetric spectral problem. Phys. Lett. A 375, 2149–2152 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kaiser, H.-C., Neidhardt, H., Rehberg, J.: Density and current of a dissipative Schrödinger operator. J. Math. Phys. 43, 5325–5350 (2002)

    Google Scholar 

  21. Kaiser, H.-C., Neidhardt, H., Rehberg, J.: Macroscopic current induced boundary conditions for Schrödinger-type operators. Integral Equ. Oper. Theory 45, 39–63 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kaiser, H.-C., Neidhardt, H., Rehberg, J.: On 1-dimensional dissipative Schrödinger-type operators their dilations and eigenfunction expansions. Mathematische Nachrichten 252, 51–69 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Karabash, I., Kostenko, A.: Similarity of \({\rm sgn} x(-\tfrac{{\rm d}^2}{{\rm d}x^2} + c\delta )\) type operators to normal and self-adjoint operators. Math. Notes 74, 127–131 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Karabash, I., Kostenko, A.: On the similarity of a \(J\) -nonnegative Sturm-Liouville operator to a self-adjoint operator. Funct. Anal. Appl. 43, 65–68 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Karabash, I.M., Kostenko, A.S., Malamud, M.M.: The similarity problem for \(J\)-nonnegative Sturm–Liouville operators. J. Differ. Equ. 246, 964–997 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kato, T.: Perturbation theory for linear operators. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  27. Kostenko, A.S.: Similarity of indefinite Sturm-Liouville operators with singular potential to a self-adjoint operator. Math. Notes 78, 134–139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Krein, M. G.: On Bari bases of a Hilbert space. Uspekhi Matematicheskikh Nauk 12, 333–341 (1957, in Russian)

    Google Scholar 

  29. Krejčiřík, D.: Calculation of the metric in the Hilbert space of a \({\cal{PT}}\)-symmetric model via the spectral theorem. J. Phys. A Math. Theoret. 41, 244012 (2008)

    Article  Google Scholar 

  30. Krejčiřík, D., Bíla, H., Znojil, M.: Closed formula for the metric in the Hilbert space of a \({\cal{PT}}\)-symmetric model. J. Phys. A Math. Gen. 39, 10143–10153 (2006)

    Article  MATH  Google Scholar 

  31. Krejčiřík, D., Siegl, P.: \({\cal{PT}}\) -symmetric models in curved manifolds. J. Phys. A Math. Theoret. 43, 485204 (2010)

    Article  Google Scholar 

  32. Kuzhel, S., Shapovalova, O., Vavrykovych, L.: On \(J\)-self-adjoint extensions of the Phillips symmetric operator. Methods Funct. Anal. Topol. 16, 333–348 (2010)

    MATH  MathSciNet  Google Scholar 

  33. Kuzhel, S., Trunk, C.: On a class of J-self-adjoint operators with empty resolvent set. J. Math. Anal. Appl. 379, 272–289 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. Langer, H., Tretter, C.: A Krein space approach to PT-symmetry. Czechoslovak J. Phys. 54, 1113–1120 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Malamud, M.: A criterion for a closed operator to be similar to a selfadjo int operator (English translation). Ukrainian Math. J. 37, 41–48 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  36. Mikhajlov, V.: Riesz bases in \({\cal{L}}_{2} (0,1)\). Soviet Math. Dokl. 3, 851–855 (1962)

    MATH  Google Scholar 

  37. Mostafazadeh, A.: Pseudo-Hermitian representation of quantum mechanics. Int. J. Geom. Methods Modern Phys. 7, 1191–1306 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Naboko, S.: Conditions for similarity to unitary and selfadjoint operators. Funct. Anal. Appl. 18, 13–22 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  39. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1. Functional Analysis. Academic Press, Dublin (1980)

    MATH  Google Scholar 

  40. Scholtz, F.G., Geyer, H.B., Hahne, F.J.W.: Quasi-Hermitian operators in quantum mechanics and the variational principle. Ann. Phys. 213, 74–101 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  41. Shkalikov, A.: On the basis property of root vectors of a perturbed self-adjoint operator. Proc. Steklov Inst. Math. 269, 284–298 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  42. Siegl, P.: Quasi-Hermitian Models. Master’s thesis, FNSPE, CTU in Prague (2007/2008)

  43. Siegl, P.: Supersymmetric quasi-Hermitian Hamiltonians with point interactions on a loop. J. Phys. A Math. Theoret. 41, 244025 (2008)

    Article  MathSciNet  Google Scholar 

  44. Siegl, P.: \({\cal{PT}}\)-symmetric square well-perturbations and the existence of metric operator. Int. J. Theoret. Phys. 50, 991–996 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  45. van Casteren, J.A.: Operators similar to unitary or selfadjoint ones. Pacific J. Math. 104, 241–255 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  46. Železný, J.: The Krein-space theory for non-Hermitian \({\cal {PT}}\)-symmetric operators. Master’s thesis, FNSPE, CTU in Prague (2010/2011)

  47. Železný, J.: Spectrum of the metric operator of a simple \({\cal{PT}}\)-symmetric model. Int. J. Theoret. Phys. 50, 1012–1018 (2011)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

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

    David Krejčiřík, Petr Siegl & Jakub Železný

  2. IKERBASQUE, Basque Foundation for Science, Alameda Urquijo, 36, 5, 48011 , Bilbao, Spain

    David Krejčiřík

  3. Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic

    Petr Siegl & Jakub Železný

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

    Petr Siegl

Authors
  1. David Krejčiřík
    View author publications

    Search author on:PubMed Google Scholar

  2. Petr Siegl
    View author publications

    Search author on:PubMed Google Scholar

  3. Jakub Železný
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to David Krejčiřík.

Additional information

Communicated by Henk de Snoo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krejčiřík, D., Siegl, P. & Železný, J. On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators. Complex Anal. Oper. Theory 8, 255–281 (2014). https://doi.org/10.1007/s11785-013-0301-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue date:

  • DOI: https://doi.org/10.1007/s11785-013-0301-y

Keywords

Mathematics Subject Classification (2010)