Skip to main content
SpringerLink
Log in
Book cover

Partial Differential Equations and Spectral Theory pp 73–117Cite as

Large Coupling Convergence: Overview and New Results

  • Chapter
  • First Online:

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 211))

Abstract

In this paper we present a couple of old and new results related to the problem of large coupling convergence. Several aspects of convergence are discussed, namely norm resolvent convergence as well as convergence within Schatten-von Neumann classes. We also discuss the rate of convergence with a special emphasis on the optimal rate of convergence, for which we give necessary and sufficient conditions. The collected results are then used for the case of Dirichlet operators. Our method is purely analytical and is supported by a wide variety of examples.

Mathematics Subject Classification (2000). Primary: 47B25; Secondary: 47A07, 47F05, 60J35.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, AMS Chelsea Publishing 2004.

    Google Scholar 

  2. A. Ben Amor, Trace inequalities for operators associated to regular Dirichlet forms. Forum Math. 16 (2004), 417–429.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Ben Amor, Sobolev-Orlicz inequalities, ultracontractivity and spectra of time changed Dirichlet forms. Math. Z. 255 (2007), 627–647.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Ben Amor and J.F. Brasche, Sharp estimates for large coupling convergence with applications to Dirichlet operators. J. Funct.Anal. 254 (2008), 454–475.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.F. Brasche, Perturbation of Schrödinger Hamiltonians by measures: Self-adjointness and lower semiboundedness. J. Math. Phys. 26 (1985), 621–626.

    Article  MATH  MathSciNet  Google Scholar 

  6. J.F. Brasche, Upper bounds for Neumann-Schatten norms. Potential Anal. 14 (2001), 175–205.

    Article  MATH  MathSciNet  Google Scholar 

  7. J.F. Brasche and M. Demuth, Dynkin’s formula and large coupling convergence. J. Funct. Anal. 219 (2005), 34–69.

    Article  MATH  MathSciNet  Google Scholar 

  8. J.F. Brasche and M. Demuth, Resolvent differences for general obstacles. In: Nonlinear elliptic and parabolic problems, Progr. Nonlinear Differential Equations Appl., vol. 64, pp. 63–70, Birkhäuser, Basel, 2005.

    Google Scholar 

  9. V. Bruneau and G. Carbou, Spectral asymptotic in the large coupling limit. Asymptot. Anal. 29 (2002), 91–113.

    MATH  MathSciNet  Google Scholar 

  10. Z.-Q. Chen, M. Fukushima and J. Ying, Traces of symmetric Markov processes and their characterizations. Ann. Prob. 34 (2006), 1052–1102.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Demuth, On large coupling operator norm convergences of resolvent differences. J. Math. Phys. 32 (1991), 1522–1530.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Demuth, W. Kirsch, and I. McGillivray, Schr¨odinger semigroups – geometric estimates in terms of the occupation time. Comm. Part. Diff. Equat. 20 (1995), 37– 57.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet forms and Symmetric Markov Processes. Walter de Gruyter, Berlin-New York, 1994.

    MATH  Google Scholar 

  14. L. Grubisić, Ritz value estimates and applications in mathematical physics. Dissertation, Fernuniversit¨at Hagen, 2005.

    Google Scholar 

  15. L. Grubisi´c, Relative convergence estimates for the spectral asymptotic in the large coupling limit. Integral Equations and Operator Theory 65 (2009), no. 1, 51–81.

    Google Scholar 

  16. M. Hansmann, Schr¨odingeroperatoren mit exakten Konvergenzraten. Diplomarbeit, TU Clausthal, 2005.

    Google Scholar 

  17. R. Hempel, Schr¨odinger operator with strong magnetic fields of compact support. Recent advances in differential equations and mathematical physics. Contemp. Math. 412 (2006), 199–205.

    MathSciNet  Google Scholar 

  18. Andrew V. Knyazev and O. Widlung, Lavrentiev regularization + Ritz approximation + uniform finite element error estimates for differential equations with rough coefficients. Math. Comp. 72 (2003), no. 241, 17–40.

    Google Scholar 

  19. Andrew V. Knyazev, Observations on degenerate saddle point problems. Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3742–3749.

    Google Scholar 

  20. Z. Ma and M. R¨ockner, Dirichlet Forms. Springer, Berlin-Heidelberg-New York, 1992.

    Google Scholar 

  21. M. Zuhair Nashed, Perturbations and approximations for generalized inverses and linear operator equations. In: Generalized inverses and applications, Proc. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1973, pp. 325–396. Publ. Math. Res. Center Univ. Wisconsin, No. 32, Academic Press, 1976.

    Google Scholar 

  22. M. Reed and B. Simon: Methods of Modern Mathematical Physics, I. Academic Press, 1980.

    Google Scholar 

  23. B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28 (1978), 377–385.

    Article  MATH  MathSciNet  Google Scholar 

  24. B. Simon: Trace ideals and their applications, volume 120 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, 2005.

    MATH  Google Scholar 

  25. P. Stollmann: St¨orungstheorie von Dirichletformen mit Anwendungen auf Schr¨odingeroperatoren. Habilitationsschrift, Frankfurt, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I.P.E.I.N, Tunis, Tunisia

    Hichem BelHadjAli

  2. Faculty of Sciences of Gafsa, Gafsa, Tunisia

    Ali Ben Amor

  3. Technische Universität Clausthal, Clausthal-Zellerfeld, Germany

    Johannes F. Brasche

Authors
  1. Hichem BelHadjAli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Ben Amor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Johannes F. Brasche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hichem BelHadjAli .

Editor information

Editors and Affiliations

  1. FB Mathematik u. Informatik, Inst. Mathematik, TU Clausthal, Erzstr. 1, Clausthal-Zellerfeld, 38678, Germany

    Michael Demuth

  2. Inst. Mathematik, Abt. Analysis, Universität Potsdam, Potsdam, Brandenburg, Germany

    Bert-Wolfgang Schulze

  3. FB Mathematik, Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, Göttingen, 37073, Germany

    Ingo Witt

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Basel

About this chapter

Cite this chapter

BelHadjAli, H., Amor, A.B., Brasche, J.F. (2011). Large Coupling Convergence: Overview and New Results. In: Demuth, M., Schulze, BW., Witt, I. (eds) Partial Differential Equations and Spectral Theory. Operator Theory: Advances and Applications(), vol 211. Springer, Basel. https://doi.org/10.1007/978-3-0348-0024-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation