Functional Analysis and Its Applications

, Volume 52, Issue 1, pp 66–69 | Cite as

Essential Spectrum of Schrödinger Operators on Periodic Graphs

  • V. S. Rabinovich
Brief Communications


We give a description of the essential spectra of unbounded operators ℋ q on L2(Γ) determined by the Schrödinger operators −d2/dx2 + q(x) on the edges of Γ and general vertex conditions. We introduce a set of limit operators of ℋ q such that the essential spectrum of ℋ q is the union of the spectra of limit operators. We apply this result to describe the essential spectra of the operators ℋ q with periodic potentials perturbed by terms slowly oscillating at infinity.

Key words

periodic graph Schrödinger operator on a graph limit operator essential spectrum 


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  1. [1]
    M. S. Agranovich, in: Partial Differential Equations VI, Encyclopaedia of Mathematical Sciences, vol. 63, 1994, 1–130.CrossRefGoogle Scholar
  2. [2]
    G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, Amer. Math. Soc., Providence, RI, 2013.Google Scholar
  3. [3]
    E. Korotyaev and I. Lobanov, Ann. Henri Puancaré, 8:6 (2007), 1151–1176.CrossRefGoogle Scholar
  4. [4]
    E. Korotyaev and N. Saburova, Scholar
  5. [5]
    P. Kuchment, Waves in Random Media, 14:1 (2004), 107–128.MathSciNetCrossRefGoogle Scholar
  6. [6]
    P. Kuchment, J. Phys. A: Math. Gen., 38:22 (2005), 4887–4900.CrossRefGoogle Scholar
  7. [7]
    P. Kuchment and O. Post, Comm. Math. Phys., 275:3 (2007), 805–826.MathSciNetCrossRefGoogle Scholar
  8. [8]
    M. Lindner and M. Seidel, J. Funct. Anal., 267:3 (2014), 901–917.MathSciNetCrossRefGoogle Scholar
  9. [9]
    V. S. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and its Applications in Operator Theory, Operator Theory: Advances and Applications, vol. 150, Birkhäuser, Basel, 2004.Google Scholar
  10. [10]
    V. S. Rabinovich, Russ. J. Math. Phys., 12:1 (2005), 62–80.MathSciNetGoogle Scholar
  11. [11]
    V. S. Rabinovich and S. Roch, J. Phys. A, Math. Theor., 39:26 (2006), 8377–8394.Google Scholar
  12. [12]
    V. S. Rabinovich and S. Roch, J. Phys. A, Math. Theor., 40:33 (2007), 10109–10128.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituto Politécnico NacionalESIME ZacatencoMexico City, the United Mexican StatesMexico

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