Eigenvalue Estimates for Bilayer Graphene

  • Jean-Claude CueninEmail author


Recently, Ferrulli–Laptev–Safronov  (2016) obtained eigenvalue estimates for an operator associated with bilayer graphene in terms of \(L^q\) norms of the (possibly non-self-adjoint) potential. They proved that for \(1<q<4/3\) all nonembedded eigenvalues lie near the edges of the spectrum of the free operator. In this note, we prove this for the larger range \(1\le q\le 3/2\). The latter is optimal if embedded eigenvalues are also considered. We prove similar estimates for a modified bilayer operator with so-called trigonal warping term. Here, the range for q is smaller since the Fermi surface has less curvature. The main tool is new uniform resolvent estimates that may be of independent interest and is collected in an appendix (in greater generality than needed).


Bilayer graphene Trigonal warping Eigenvalue estimates Complex potentials Embedded eigenvalues 

Mathematics Subject Classification



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  1. 1.
    Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Cuenin, J.-C.: Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials. J. Funct. Anal. 272(7), 2987–3018 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cuenin, J.-C.: Embedded eigenvalues for generalized Schrödinger operators, p. 18 (2017). ArXiv:1709.06989
  4. 4.
    Cuenin, J.-C., Laptev, A., Tretter, C.: Eigenvalue estimates for non-selfadjoint Dirac operators on the real line. Ann. Henri Poincaré 15(4), 707–736 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cuenin, J.-C., Siegl, P.: Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications. Lett. Math. Phys. 108(7), 1757–1778 (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ferrulli, F., Laptev, A., Safronov, O.: Eigenvalues of the bilayer graphene operator with a complex valued potential. ArXiv e-prints (2016)Google Scholar
  7. 7.
    Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials. Bull. Lond. Math. Soc. 43(4), 745–750 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frank, R.L.: Eigenvalue bounds for Schrödinger operators with complex potentials III. Trans. Am. Math. Soc. 370(1), 219–240 (2018)CrossRefzbMATHGoogle Scholar
  9. 9.
    Frank, R.L., Sabin, J.: Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates. Am. J. Math. 139(6), 1649–1691 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frank, R.L., Simon, B.: Eigenvalue bounds for Schrödinger operators with complex potentials II. J. Spectr. Theory 7(3), 633–658 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gesztesy, F., Latushkin, Y., Mitrea, M., Zinchenko, M.: Nonselfadjoint operators, infinite determinants, and some applications. Russ. J. Math. Phys. 12(4), 443–471 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Greenleaf, A.: Principal curvature and harmonic analysis. Indiana Univ. Math. J. 30(4), 519–537 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hörmander, L.: The analysis of linear partial differential operators I, volume 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Distribution Theory and Fourier Analysis, 2nd edn. Springer, Berlin (1990)Google Scholar
  14. 14.
    Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)Google Scholar
  15. 15.
    Katsnelson, M.I.: Graphene: Carbon in Two Dimensions. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  16. 16.
    Kenig, C.E., Ruiz, A., Sogge, C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Laptev, A., Safronov, O.: Eigenvalue estimates for Schrödinger operators with complex potentials. Commun. Math. Phys. 292(1), 29–54 (2009)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Littman, W.: Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Am. Math. Soc. 69, 766–770 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Riss, U.V., Meyer, H.-D.: Calculation of resonance energies and widths using the complex absorbing potential method. J. Phys. B 26(23), 4503–4535 (1993)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Simon, B.: Trace Ideals and Their Applications, Volume 120 of Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence, RI (2005)Google Scholar
  21. 21.
    Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. With the Assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ (1993)Google Scholar
  22. 22.
    Vozmediano, M.A., Katsnelson, M., Guinea, F.: Gauge fields in graphene. Phys. Rep. 496(4), 109–148 (2010)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Zworski, M.: Scattering resonances as viscosity limits (2015). ArXiv e-prints arXiv:1505.00721

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Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMunichGermany

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