Skip to main content
SpringerLink
Log in

Numerical Illustrations

  • Chapter
  • First Online:
Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Part of the book series: Pseudo-Differential Operators ((PDO,volume 14))

  • 690 Accesses

Abstract

In this chapter we give some numerical illustrations to the results about the asymptotic distribution of eigenvalues. Such calculations have already been carried out by many people, Trefethen, Trefethen and Embree, Davies, Davies and Hager, Zworski and many others. Numerical calculations with special attention to Weyl asymptotics have been carried out by Hager, Bordeaux Montrieux, Vogel. Many of the illustrations below are therefore well-known and even though we wrote our own Matlab programs, we have clearly benefitted from the preceding works.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    agnes.sjo@gmail.com

References

  1. W. Bordeaux Montrieux, Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, thèse, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367

  2. E.B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators. Commun. Math. Phys. 200(1), 35–41 (1999)

    Article  Google Scholar 

  3. E.B. Davies, Pseudopectra of differential operators. J. Operator Theory 43, 243–262 (2000)

    MathSciNet  Google Scholar 

  4. E.B. Davies, M. Hager, Perturbations of Jordan matrices. J. Approx. Theory 156(1), 82–94 (2009)

    Article  MathSciNet  Google Scholar 

  5. M. Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle. Ann. Fac. Sci. Toulouse Math. 15(2), 243–280 (2006)

    Article  MathSciNet  Google Scholar 

  6. M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré 7(6), 1035–1064 (2006)

    Article  MathSciNet  Google Scholar 

  7. M. Hitrik, Boundary spectral behavior for semiclassical operators in dimension one. Int. Math. Res. Not. 2004(64), 3417–3438 (2004)

    Article  MathSciNet  Google Scholar 

  8. M. Hitrik, J. Sjöstrand, Rational invariant tori and band edge spectra for non-selfadjoint operators. J. Eur. Math. Soc. 20(2), 391–457 (2018). http://arxiv.org/abs/1502.06138

    Article  MathSciNet  Google Scholar 

  9. J. Sjöstrand, M. Vogel, Interior eigenvalue density of large bi-diagonal matrices subject to random perturbations, in Proceedings of the Conference “Microlocal Analysis and Singular Perturbation Theory”, RIMS, Kyoto University, October 5–9, 2015, RIMS Conference Proceedings series “RIMS Kokyuroku Bessatsu” (2015). http://arxiv.org/abs/1604.05558

  10. J. Sjöstrand, M. Vogel, Large bi-diagonal matrices and random perturbations. J. Spectr. Theory 6(4), 977–1020 (2016). http://arxiv.org/abs/1512.06076

    Article  MathSciNet  Google Scholar 

  11. J. Sjöstrand, M. Vogel, Interior eigenvalue density of Jordan matrices with random perturbations. In Analysis Meets Geometry: A Tribute to Mikael Passare. Trends in Mathematics, pp. 439–466 (Springer, Cham, 2017). http://arxiv.org/abs/1412.2230

    Google Scholar 

  12. L.N. Trefethen, Pseudospectra of linear operators. SIAM Rev. 39(3), 383–406 (1997)

    Article  MathSciNet  Google Scholar 

  13. L.N. Trefethen, M. Embree, Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, 2005)

    Google Scholar 

  14. M. Vogel, The precise shape of the eigenvalue intensity for a class of non-selfadjoint operators under random perturbations. Ann. Henri Poincaré 18(2), 435–517 (2017). http://arxiv.org/abs/1401.8134

    Article  MathSciNet  Google Scholar 

  15. M. Vogel, Two point eigenvalue correlation for a class of non-selfadjoint operators under random perturbations. Commun. Math. Phys. 350(1), 31–78 (2017). http://arxiv.org/abs/1412.0414

    Article  MathSciNet  Google Scholar 

  16. M. Zworski, A remark on a paper of E. B. Davies: “Semi-classical states for non-self-adjoint Schrödinger operators”. Proc. Am. Math. Soc. 129(10), 2955–2957 (2001)

    Google Scholar 

  17. M. Zworski, Numerical linear algebra and solvability of partial differential equations. Commun. Math. Phys. 229(2), 293–307 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Bourgogne Franche-Comté, Dijon, France

    Johannes Sjöstrand

Authors
  1. Johannes Sjöstrand
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Sjöstrand, J. (2019). Numerical Illustrations. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_20

Download citation

Publish with us

Policies and ethics

Search

Navigation