Annales Henri Poincaré

, Volume 19, Issue 11, pp 3397–3455 | Cite as

Inverse Scattering for Schrödinger Operators on Perturbed Lattices

  • Kazunori Ando
  • Hiroshi IsozakiEmail author
  • Hisashi Morioka


We study the inverse scattering for Schrödinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa 2, 151–218 (1975)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agmon, S., Hörmander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. d’Anal. Math. 30, 1–38 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ando, K.: Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice. Ann. Henri Poincaré 14, 347–383 (2013)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ando, K., Isozaki, H., Morioka, H.: Spectral properties of Schrödinger operators on perturbed lattices. Ann. Henri Poincaré 17, 2103–2171 (2016)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Burago, D., Ivanov, S., Kurylev, Y.: A graph discretization of the Laplace–Beltrami operator. J. Spectr. Theory 4, 675–714 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Calderón, A.P.: On an inverse boundary value problem. In: Meyer, W.H., Raupp, M.A. (eds.) Seminar on Numerical Analysis and its Applications to Continuum Physics, pp. 65–73. Sociedade Brasileira de Matematica, Rio de Janeiro (1980)Google Scholar
  7. 7.
    Chadan, K., Colton, D., Päivärinta, L., Rundell, W.: An Introduction to Inverse Scattering and Inverse Boundary Value Problems. SIAM, Philadelphia (1997)CrossRefGoogle Scholar
  8. 8.
    Chung, R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  9. 9.
    Colin de Verdière, Y.: Réseaux électriques planaires I. Comment. Math. Helv. 69, 351–374 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Colin de Verdière, Y.: Cours Spécialisés 4, Spectre de Graphes. Soc. Math. de France (1998)Google Scholar
  11. 11.
    Colin de Verdière, Y., Françoise, T.: Scattering theory for graphs isomorphic to a regular tree at infinity. J. Math. Phys. 54, 063502 (2013)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Colin de Verdière, Y., de Gitler, I., Vertigan, D.: Réseauxélectriques planaires II. Comment. Math. Helv. 71, 144–167 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cuenin, J.C., Siedentop, H.: Dipoles in graphene have infinitely many bound states. J. Math. Phys. 55, 122304 (2014)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Curtis, E.B., Morrow, J.A.: Determining the resistors in a network. SIAM J. Appl. Math. 50, 918–930 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Curtis, E.B., Morrow, J.A.: The Dirichlet to Neumann map for a resistor network. SIAM J. Appl. Math. 51, 1011–1029 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Curtis, E.B., Morrow, J.A.: Inverse Problems for Electrical Networks. World Scientific, Singapore (2000)CrossRefGoogle Scholar
  17. 17.
    Curtis, E.B., Mooers, E., Morrow, J.A.: Finding the conductors in circular networks. Math. Model. Numer. Anal. 28, 781–813 (1994)CrossRefGoogle Scholar
  18. 18.
    Curtis, E.B., Ingerman, D., Morrow, J.A.: Circular planar graphs and resistor networks. Linear Algbr. Its Appl. 283, 115–150 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284, 787–794 (1984)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Eskina, M.S.: The direct and the inverse scattering problem for a partial difference equation. Sov. Math. Dokl. 7, 193–197 (1966)MathSciNetGoogle Scholar
  21. 21.
    Faddeev, L.D.: Uniqueness of the inverse scattering problem. Vestnik Leningrad Univ. 11, 126–130 (1956)MathSciNetGoogle Scholar
  22. 22.
    Faddeev, L.D.: Increasing solutions of the Schrödinger equations. Sov. Phys. Dokl. 10, 1033–1035 (1966)ADSGoogle Scholar
  23. 23.
    Faddeev, L.D.: Inverse problem of quantum scattering theory. J. Sov. Math. 5, 334–396 (1976)CrossRefGoogle Scholar
  24. 24.
    Gérard, C., Nier, F.: The Mourre theory for analytically fibred operators. J. Funct. Anal. 152, 202–219 (1989)CrossRefGoogle Scholar
  25. 25.
    González, J., Guiner, F., Vozmediano, M.A.H.: The electronic spectrum of fullerenes from the Dirac equation. Nucl. Phys. B 406, 771–794 (1993)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Higuchi, Y., Shirai, T.: Some spectral and geometric properties for infinite graphs. Contemp. Math. 347, 29–56 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Higuchi, Y., Nomura, Y.: Spectral structure of the Laplacian on a covering graph. Euro. J. Comb. 30, 570–585 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hiroshima, F., Sakai, I., Shirai, T., Suzuki, A.: Note on the spectrum of discrete Schrödinger operators. J. Math. Ind. 4, 105–108 (2012)zbMATHGoogle Scholar
  29. 29.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III, Pseudodifferential Operators. Springer, Berlin (1985)zbMATHGoogle Scholar
  30. 30.
    Ikehata, M.: Reconstruction of obstacles from boundary measurements. Wave Motion 30, 205–223 (1999)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ikehata, M.: Inverse scattering problems and the enclosure method. Inverse Probl. 20, 533–551 (2004)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Isozaki, H.: Inverse spectral theory. In: Araki, H., Ezawa, H. (eds.) Topics in the Theory of Schrödinger Operators, pp. 93–143. World Scientific, Singapore (2003)Google Scholar
  33. 33.
    Isozaki, H., Korotyaev, E.: Inverse problems, trace formulae for discrete Schrödinger operators. Ann. Henri Poincaré 13, 751–788 (2012)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Isozaki, H., Morioka, H.: A Rellich type theorem for discrete Schrödinger operators. Inverse Probl. Imaging 8, 475–489 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Isozaki, H., Morioka, H.: Inverse scattering at a fixed energy for discrete Schrödinger operators on the square lattice. Ann. Inst. Fourier 65, 1153–1200 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kuroda, S.T.: Scattering theory for differential operators, I, II. J. Math. Soc. Jpn. 25(75–104), 222–234 (1973)CrossRefGoogle Scholar
  37. 37.
    Khenkin, G.M., Novikov, R.G.: The \(\overline{\partial }\)-equation in the multi-dimensional inverse scattering problem. Russ. Math. Surv. 42, 109–180 (1987)CrossRefGoogle Scholar
  38. 38.
    Kobayashi, T., Ono, K., Sunada, T.: Periodic Schrödinger operators on a manifold. Forum Math. 1, 69–79 (1989)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kondo, T., Casolo, S., Suzuki, T., Shikano, T., Sakurai, M., Harada, Y., Saito, M., Oshima, M., Trioni, M., Tantardini, G., Nakamura, J.: Atomic-scale characterization of nitrogen-doped graphite: effects of dopant nitrogen on the local electronic structure of the surrounding carbon atoms. Phys. Rev. B 86, 035436 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Korotyaev, E., Kutsenko, A.: Zigzag nanoribbons in external electric fields. Asymptot. Anal. 66, 187–206 (2010)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Korotyaev, E., Saburova, N.: Schrödinger operators on periodic graphs. J. Math. Anal. Appl. 420, 576–611 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Kotani, M., Shirai, T., Sunada, T.: Asymptotic behavior of the transition probability of a random walk on an infinite graph. J. Funct. Anal. 159, 664–689 (1998)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275, 805–826 (2007)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21, 209–234 (1989)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Muhometov, R.G.: The problem of recovery of a two-dimensional Riemannian metric and integral geometry. Soviet Math. Dokl. 18, 27–31 (1977)Google Scholar
  46. 46.
    Nachman, A.: Reconstruction from boundary measurements. Ann. Math. 128, 531–576 (1988)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Nachman, A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. 143, 71–96 (1996)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Nakamura, S.: Modified wave operators for discrete Schrödinger operators with long-range perturbations. J. Math. Phys. 55, 112101 (2014)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Neto, A.H.C., Guiner, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)ADSCrossRefGoogle Scholar
  50. 50.
    Novikov, R.G.: A multidimensional inverse spectral problem for the equation \(- \Delta \psi + (v(x)-E)\psi =0\). Funct. Anal. Appl. 22, 263–272 (1988)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Oberlin, R.: Discrete inverse problems for Schrödinger and resistor networks. Research archive of Research Experiences for Undergraduates program at University of Washington (2000)Google Scholar
  52. 52.
    Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 161, 1093–1110 (2005)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Shaban, W., Vainberg, B.: Radiation conditions for the difference Schrödinger operators. Appl. Anal. 80, 525–556 (2001)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Shipman, S.P.: Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators. Commun. Math. Phys. 332, 605–626 (2014)ADSMathSciNetCrossRefGoogle Scholar
  55. 55.
    Shirai, T.: The spectrum of infinite regular line graphs. Trans. Am. Math. Soc. 352, 115–132 (1999)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Sunada, T.: A periodic Schrödinger operator on abelian cover. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 575–583 (1990)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Suzuki, A.: Spectrum of the Laplacian on a covering graph with pendant edges: the one-dimensional lattice and beyond. Linear Algebra Appl. 439, 3464–3489 (2013)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Probl. 25, 123011 (2009)ADSCrossRefGoogle Scholar
  60. 60.
    Yafaev, D.: Mathematical Scattering Theory. American Mathematical Society, Providence (2009)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Kazunori Ando
    • 1
  • Hiroshi Isozaki
    • 2
    Email author
  • Hisashi Morioka
    • 3
  1. 1.Department of Electrical and Electronic Engineering and Computer ScienceEhime UniversityMatsuyamaJapan
  2. 2.Professor EmeritusUniversity of TsukubaTsukubaJapan
  3. 3.Faculty of Science and EngineeringDoshisha UniversityKyotanabeJapan

Personalised recommendations