Skip to main content
SpringerLink
Log in

On the Ground State for the NLS Equation on a General Graph

  • Chapter
  • First Online:
Advances in Quantum Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 18))

  • 1065 Accesses

Abstract

We review some recent results on the existence of the ground state for a nonlinear Schrödinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. In particular we concentrate on the main theorem in Cacciapuoti et al. (Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, preprint arXiv:1608.01506) which covers the most general setting and we compare it with similar results.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and 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

Institutional subscriptions

References

  1. R. Adami, D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ′ interaction. Commun. Math. Phys. 318, 247–289 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Fast solitons on star graphs. Rev. Math. Phys. 23(4), 409–451 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Adami, C. Cacciapuoti, D. Finco, D. Noja, On the structure of critical energy levels for the cubic focusing NLS on star graphs. J. Phys. A Math. Theor. 45, 192001, 7pp. (2012)

    Google Scholar 

  4. R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Stationary states of NLS on star graphs. Europhys. Lett. 100, 10003, 6pp. (2012)

    Google Scholar 

  5. R. Adami, D. Noja, N. Visciglia, Constrained energy minimization and ground states for NLS with point defects. Discrete Contin. Dyn. Syst. B 18, 1155–1188 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph. J. Differ. Equ. 257, 3738–3777 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. Poincaré Anal. Non Linear 31(6), 1289–1310 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Adami, E. Serra, P. Tilli, NLS ground states on graphs. Calc. Var. Partial Differ. Equ. 54(1), 743–761 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Adami, E. Serra, P. Tilli, Lack of Ground State for NLSE on Bridge-Type Graphs. Springer Proceedings in Mathematics and Statistics, vol. 128 (Springer, Berlin 2015)

    Google Scholar 

  10. R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy. J. Differ. Equ. 260, 7397–7415 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Adami, E. Serra, P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs. J. Funct. Anal. 271(1), 201–223 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Ali Mehmeti, K. Ammari, S. Nicaise, Dispersive effects for the Schrödinger equation on a tadpole graph. J. Math. Anal. Appl. 448, 262–280 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Banica, L.I. Ignat, Dispersion for the Schrödinger equation on networks. J. Math. Phys. 52(8), 083703, 14pp. (2011)

    Google Scholar 

  14. V. Banica, L.I. Ignat, Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees. Anal. Partial Differ. Equ. 7(4), 903–927 (2014)

    MATH  Google Scholar 

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

    Google Scholar 

  16. G. Berkolaiko, W. Liu, Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph. J. Math. Anal. Appl. 445, 803–818 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Cacciapuoti, D. Finco, D. Noja, Topology induced bifurcations for the NLS on the tadpole graph. Phys. Rev. E 91(1), 013206, 8 pp. (2015)

    Google Scholar 

  18. C. Cacciapuoti, D. Finco, D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials (2016). preprint arXiv:1608.01506

    Google Scholar 

  19. T. Cazenave, P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)

    Article  MATH  Google Scholar 

  20. P. Exner, M. Jex, On the ground state of quantum graphs with attractive δ-coupling. Phys. Lett. A 376, 713–717 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Exner, O. Turek, Approximations of singular vertex couplings in quantum graphs. Rev. Math. Phys. 19, 571–606 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Gnutzman, D. Walter, Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory. Phys. Rev. E 93, 032204 (2016)

    Article  MathSciNet  Google Scholar 

  23. S. Gnutzman, D. Walter, Stationary waves on nonlinear quantum graphs II: application of canonical perturbation theory in basic graph structures. Phys. Rev. E 94, 062216 (2016)

    Article  Google Scholar 

  24. S. Gnutzman, U. Smilansky, S. Derevyanko, Stationary scattering from a nonlinear network. Phys. Rev. A 83, 033831 (2011)

    Article  Google Scholar 

  25. S. Haeseler, Heat kernel estimates and related inequalities on metric graphs. arXiv:1101.3010v1 (2011)

    Google Scholar 

  26. M. Keller, D. Lenz, R. Wojciechowski, Note on basic features of large time behaviour of heat kernels. J. Reine Angew. Math. 708, 73–95 (2015)

    MathSciNet  MATH  Google Scholar 

  27. E. Kirr, P.G. Kevrekidis, D.E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. Commun. Math. Phys. 308, 795–844 (2011)

    Article  MATH  Google Scholar 

  28. V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires. J. Phys. A Math. Gen. 32(4), 595–630 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. J. Marzuola, D. E. Pelinovsky, Ground states on the dumbbell graph. Appl. Math. Res. Exp. 2016, 98–145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Mugnolo, Semigroup Methods for Evolution Equations on Networks (Springer, New York, 2014)

    Book  MATH  Google Scholar 

  31. D. Noja, Nonlinear Schrödinger equations on graphs: recent results and open problems. Philos. Trans. R. Soc. A 372, 20130002, 20 pp. (2014)

    Google Scholar 

  32. D. Noja, D. Pelinovsky, G. Shaikhova, Bifurcation and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph. Nonlinearity 28, 2343–2378 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Pelinovsky, G. Schneider, Bifurcations of standing localized waves on periodic graphs. Ann. Henri Poincaré 18, 1185 (2017). doi:10.1007/s00023-016-0536-z

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Reed, B. Simon, Methods of Modern Mathematical Physics IV, Analysis of Operators (Academic, London, 1978)

    MATH  Google Scholar 

  35. A. Soffer, M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Soffer, M.I. Weinstein, Selection of the ground state for nonlinear Schroedinger equations. Rev. Math. Phys. 16, 977–1071 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facoltà di Ingegneria, Università Telematica Internazionale Uninettuno, Corso Vittorio Emanuele II 39, 00186, Roma, Italy

    Domenico Finco

Authors
  1. Domenico Finco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Domenico Finco .

Editor information

Editors and Affiliations

  1. International Sch. for Advanced Studies, SISSA, Trieste, Italy

    Alessandro Michelangeli

  2. International Sch. for Advanced Studies, SISSA and Sapienza University of Rome, Trieste, Italy

    Gianfausto Dell'Antonio

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Finco, D. (2017). On the Ground State for the NLS Equation on a General Graph. In: Michelangeli, A., Dell'Antonio, G. (eds) Advances in Quantum Mechanics. Springer INdAM Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-58904-6_9

Download citation

Publish with us

Policies and ethics

Search

Navigation