Skip to main content
SpringerLink
Log in

Lieb–Thirring Type Inequalities for Schrödinger Operators with a Complex-Valued Potential

  • Open Problems
  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

We present a possible generalization of the Lieb–Thirring inequalities for eigenvalues of Schrödinger operators to the case of complex potentials. We ask for a proof or disproof of this generalization. Some weaker results which have been obtained are reviewed.

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. Abramov A.A., Aslanyan A., Davies E.B.: Bounds on complex eigenvalues and resonances. J. Phys. A 34(1), 57–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruneau V., Ouhabaz E.M.: Lieb–Thirring estimates for non-self-adjoint Schrödinger operators. J. Math. Phys. 49(9), 093504 (2008)

    Article  MathSciNet  Google Scholar 

  3. Demuth M., Hansmann M., Katriel G.: On the discrete spectrum of non-selfadjoint operators. J. Funct. Anal. 257(9), 2742–2759 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Demuth, M., Hansmann, M., Katriel, G.: Eigenvalues of non-selfadjoint operators: a comparison of two approaches. In: Proceedings of the conference “Mathematical Physics, Spectral Theory and Stochastic Analysis” in Goslar, September 11–16 (2012, in press)

  5. Frank R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Frank R.L., Laptev A., Lieb E.H., Seiringer R.: Lieb–Thirring inequalities for Schrödinger operators with complex-valued potentials. Lett. Math. Phys. 77(3), 309–316 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hansmann M.: An eigenvalue estimate and its application to non-selfadjoint Jacobi and Schrödinger operators. Lett. Math. Phys. 98(1), 79–95 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hundertmark, D.: Some bound state problems in quantum mechanics. In: Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. Proceedings of Symposium Pure Mathematics, vol. 76, pp. 463–496. American Mathematical Society, Providence (2007)

  9. Laptev A., Safronov O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Comm. Math. Phys. 292(1), 29–54 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lieb E.H., Thirring W.: Bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett. 35, 687–689 (1975)

    Article  Google Scholar 

  11. Pavlov, B.S.: On a non-selfadjoint Schrödinger operator. In: Problems of Mathematical Physics, No. I. Spectral Theory and Wave Processes (Russian), pp. 102–132. Izdat. Leningrad. Univ., Leningrad (1966)

  12. Pavlov, B.S.: On a non-selfadjoint Schrödinger operator. II. In: Problems of Mathematical Physics, No. 2. Spectral Theory, Diffraction Problems (Russian), pp. 133–157. Izdat. Leningrad. Univ., Leningrad (1967)

  13. Pavlov, B.S.: On a nonselfadjoint Schrödinger operator. III. In: Problems of Mathematical Physics, No. 3. Spectral theory (Russian), pp. 59–80. Izdat. Leningrad. Univ., Leningrad (1968)

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematics, Clausthal University of Technology, Clausthal, Germany

    Michael Demuth

  2. Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany

    Marcel Hansmann

  3. Department of Mathematics, ORT Braude College, Karmiel, Israel

    Guy Katriel

Authors
  1. Michael Demuth
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel Hansmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guy Katriel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guy Katriel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demuth, M., Hansmann, M. & Katriel, G. Lieb–Thirring Type Inequalities for Schrödinger Operators with a Complex-Valued Potential. Integr. Equ. Oper. Theory 75, 1–5 (2013). https://doi.org/10.1007/s00020-012-2021-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-012-2021-5

Keywords

Search

Navigation