Skip to main content
SpringerLink
Log in
Book cover

Conference on Semigroups of Operators: Theory and Applications

SOTA 2018: Semigroups of Operators – Theory and Applications pp 47–67Cite as

Laplacians with Point Interactions—Expected and Unexpected Spectral Properties

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 325))

Abstract

We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line \([0,\infty )\). All possible boundary conditions that define generators of \(C_0\)-semigroups on \(L^2\big ([0,\infty )\big )\oplus L^2\big ([0,\infty )\big )\) are characterized. Here, the Cayley transform of the matrices that describe the boundary conditions plays an important role and using an explicit representation of the Green’s functions, it allows us to study invariance properties of semigroups.

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   149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   199.99
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

References

  1. Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)

    Google Scholar 

  2. Birkhoff, G.D.: Boundary value and expansion problems of ordinary linear differential equations. Trans. Am. Math. Soc. 9(4), 373–395 (1908)

    Article  MathSciNet  Google Scholar 

  3. Birkhoff, G.D.: On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Am. Math. Soc. 9(2), 219–231 (1908)

    Article  MathSciNet  Google Scholar 

  4. Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186. American Mathematical Society, Providence, RI (2013)

    Google Scholar 

  5. Cardanobile, S., Mugnolo, D.: Parabolic systems with coupled boundary conditions. J. Differ. Equ. 247, 1229–1248 (2009)

    Article  MathSciNet  Google Scholar 

  6. Daners, D., Glück, J., Kennedy, J.B.: Eventually and asymptotically positive semigroups on Banach lattices. J. Differ. Equ. 261(5), 2607–2649 (2016)

    Article  MathSciNet  Google Scholar 

  7. Dunford, N., Schwartz, J.T.: Linear Operators. Part III: Spectral Operators. Interscience Publishers [Wiley], New York-London-Sydney (1971)

    Google Scholar 

  8. Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1987). Oxford Science Publications

    Google Scholar 

  9. Engel, K.-J., Kramar Fijavž, M.: Waves and diffusion on metric graphs with general vertex conditions. Evol. Equ. Control Theory 8, 633–661 (2019)

    Google Scholar 

  10. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)

    Google Scholar 

  11. Gomilko, A.M.: Conditions on the generator of a uniformly bounded \(C_0\)-semigroup. Funct. Anal. Appl. 33, 294–296 (1999)

    Article  MathSciNet  Google Scholar 

  12. Hussein, A., Krejčiřík, D., Siegl, P.: Non-self-adjoint graphs. Trans. Am. Math. Soc. 367(4), 2921–2957 (2015)

    Article  MathSciNet  Google Scholar 

  13. Hussein, A.: Maximal quasi-accretive Laplacians on finite metric graphs. J. Evol. Equ. 14(2), 477–497 (2014)

    Article  MathSciNet  Google Scholar 

  14. Kant, U., Klauß, T., Voigt, J., Weber, M.: Dirichlet forms for singular one-dimensional operators and on graphs. J. Evol. Equ. 9, 637–659 (2009)

    Article  MathSciNet  Google Scholar 

  15. Kostrykin, V., Potthoff, J., Schrader, R.: Contraction semigroups on metric graphs. In: Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, vol. 77, pp. 423–458. American Mathematical Society, Providence, RI (2008)

    Google Scholar 

  16. Kostrykin, V., Schrader, R.: Laplacians on metric graphs: eigenvalues, resolvents and semigroups. In: Quantum Graphs and Their Applications, Contemporary Mathematics, vol. 415, pp. 201–225. American Mathematical Society, Providence, RI (2006)

    Google Scholar 

  17. Mugnolo, D., Nittka, R.: Properties of representations of operators acting between spaces of vector-valued functions. Positivity 15, 135–154 (2011)

    Article  MathSciNet  Google Scholar 

  18. Mugnolo, D.: Gaussian estimates for a heat equation on a network. Netw. Het. Media 2, 55–79 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Mugnolo, D.: Vector-valued heat equations and networks with coupled dynamic boundary conditions. Adv. Diff. Equ. 15, 1125–1160 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Mugnolo, D.: Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer, Cham (2014)

    Google Scholar 

  21. Ouhabaz, E.M.: Invariance of closed convex sets and domination criteria for semigroups. Potential Anal. 5, 611–625 (1996)

    Article  MathSciNet  Google Scholar 

  22. Ouhabaz, E.M.: Analysis of Heat Equations on Domains. London Mathematical Society Monograph Series, vol. 30. Princeton University Press, Princeton, NJ (2005)

    Google Scholar 

  23. Shi, D.H., Feng, D.X.: Characteristic conditions of the generation of \(C_0\) semigroups in a Hilbert space. J. Math. Anal. Appl. 247, 356–376 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second author would like to thank Jochen Glück (Passau) for helpful discussions.

D.M. was partially supported by the Deutsche Forschungsgemeinschaft (Grant 397230547).

Author information

Authors and Affiliations

  1. Department of Mathematics, TU Kaiserslautern, Paul-Ehrlich-Straße, 67663, Kaiserslautern, Germany

    Amru Hussein

  2. Chair of Analysis, Faculty of Mathematics and Computer Science, FernUniversität Hagen, Hagen, Germany

    Delio Mugnolo

Authors
  1. Amru Hussein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Delio Mugnolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delio Mugnolo .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, South Africa

    Jacek Banasiak

  2. Department of Mathematics, Lublin University of Technology, Lublin, Poland

    Adam Bobrowski

  3. Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

    Mirosław Lachowicz

  4. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    Yuri Tomilov

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hussein, A., Mugnolo, D. (2020). Laplacians with Point Interactions—Expected and Unexpected Spectral Properties. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation