Skip to main content
SpringerLink
Log in

On the Point Spectrum in the Ekman Boundary Layer Problem

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

New eigenvalue enclosures for the block operator problem arising in the study of stability of the Ekman boundary layer are proved. This solves an open problem in [19] on the existence of open sets of eigenvalues in domains of Fredholmness of the analyzed operator family.

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

Fig. 1

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 Math. Gen. 34, 57 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bögli, S., Marletta, M., Tretter, C.: The essential numerical range for unbounded linear operators. J. Funct. Anal. 279, 108509 (2020)

    Article  MathSciNet  Google Scholar 

  3. Boulton, L., Krejčiřík, D., Siegl, P.: Non-self-adjoint portal. http://nsa.fjfi.cvut.cz

  4. Cassano, B., Ibrogimov, O.O., Krejčiřík, D., Štampach, F.: Location of eigenvalues of non-self-adjoint discrete dirac operators. Ann. Henri Poincaré 21, 2193–2217 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  5. Conway, J.B.: Functions of One Complex Variable, 2nd edn. Springer, New York-Berlin (1978)

    Book  Google Scholar 

  6. Cossetti, L.: Bounds on eigenvalues of perturbed Lamé operators with complex potentials. Math. Eng. 4, 1–29 (2021)

    Article  MathSciNet  Google Scholar 

  7. Cuenin, J.-C.: Estimates on complex eigenvalues for Dirac operators on the half-line. Integral Equ. Oper. Theory 79, 377–388 (2014)

    Article  MathSciNet  Google Scholar 

  8. Cuenin, J.-C.: Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials. J. Funct. Anal. 272, 2987–3018 (2017)

    Article  MathSciNet  Google Scholar 

  9. Cuenin, J.-C.: Sharp spectral estimates for the perturbed landau hamiltonian with \(L^p\) potentials. Integral Equ. Oper. Theory 88, 127–141 (2017)

    Article  MathSciNet  Google Scholar 

  10. Cuenin, J.-C.: Improved eigenvalue bounds for Schrödinger operators with slowly decaying potentials. Commun. Math. Phys. 376, 2147–2160 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  11. Cuenin, J.-C., Laptev, A., Tretter, C.: Eigenvalue estimates for non-selfadjoint dirac operators on the real line. Ann. Henri Poincaré 15, 707–736 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  12. Davies, E., Nath, J.: Schrödinger operators with slowly decaying potentials. J. Comput. Appl. Math. 148, 1–28 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  13. Faller, A.J.: An experimental study of the instability of the laminar Ekman boundary layer. J. Fluid Mech. 15, 560–576 (1963)

    Article  ADS  Google Scholar 

  14. Fanelli, L., Krejčiřík, D., Vega, L.: Spectral stability of Schrödinger operators with subordinated complex potentials. J. Spectr. Theory 8, 575–604 (2018)

    Article  MathSciNet  Google Scholar 

  15. Frank, R., Simon, B.: Eigenvalue bounds for Schrödinger operators with complex potentials. II. J. Spectr. Theory 7, 633–658 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. III. Trans. Am. Math. Soc. 370, 219–240 (2017)

    Article  Google Scholar 

  18. Gesztesy, F., Latushkin, Y., Mitrea, M., Zinchenko, M.: Nonselfadjoint operators, infinite determinants, and some applications. Russ. J. Math. Phys. 12, 443–471 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Greenberg, L., Marletta, M.: The Ekman flow and related problems: spectral theory and numerical analysis. Math. Proc. Cambridge Philos. Soc. 136, 719–764 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  20. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)

    Book  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. Lilly, D.K.: On the instability of Ekman boundary flow. J. Atmos. Sci. 23, 481–494 (1966)

    Article  ADS  Google Scholar 

  23. Marletta, M., Tretter, C.: Essential spectra of coupled systems of differential equations and applications in hydrodynamics. J. Differ. Equ. 243, 36–69 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  24. Spooner, G.F.: Continuous temporal eigenvalue spectrum of an Ekman boundary layer. Phys. Fluids 25, 1958–1963 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  25. Vladimirov, V.S.: Equations of Mathematical Physics. Marcel Dekker Inc, New York (1971)

    MATH  Google Scholar 

  26. Vladimirov, V.S.: Methods of the Theory of Generalized Functions. Taylor & Francis, London (2002)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Borbala Gerhat

  2. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00, Prague, Czech Republic

    Borbala Gerhat

  3. Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093, Zürich, Switzerland

    Orif O. Ibrogimov

  4. School of Mathematics and Physics, Queen’s University Belfast, University Road, Belfast, BT7 1NN, UK

    Petr Siegl

  5. Institute of Applied Mathematics, Graz University of Technology, Steyrergasse 30, 8010, Graz, Austria

    Petr Siegl

Authors
  1. Borbala Gerhat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Orif O. Ibrogimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Petr Siegl
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

B.G. expresses her gratitude to her supervisor C. Tretter for drawing her attention to this problem and to the Queen’s University Belfast for their hospitality during a SEMP doctoral student exchange in 2019/20. O.I. is grateful to G.M. Graf for fruitful discussions and thanks the Institute for Theoretical Physics at ETH Zurich for the kind hospitality through a postdoctoral researcher position. P.S. acknowledges the support and hospitality of the Institute for Theoretical Physics, ETH Zurich, during his visit there in July 2019 when this project was initiated.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gerhat, B., Ibrogimov, O.O. & Siegl, P. On the Point Spectrum in the Ekman Boundary Layer Problem. Commun. Math. Phys. 392, 377–397 (2022). https://doi.org/10.1007/s00220-022-04321-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-022-04321-0

Search

Navigation