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Inverse Scattering for Schrödinger Operators on Perturbed Lattices

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A Correction to this article was published on 17 December 2018

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Abstract

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).

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Change history

  • 17 December 2018

    The assumption (A-4) in p. 3406 should read as follows.

  • 17 December 2018

    The assumption (A-4) in p. 3406 should read as follows.

References

  1. Agmon, S.: Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa 2, 151–218 (1975)

    MathSciNet  MATH  Google Scholar 

  2. Agmon, S., Hörmander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. d’Anal. Math. 30, 1–38 (1976)

    Article  MathSciNet  Google Scholar 

  3. Ando, K.: Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice. Ann. Henri Poincaré 14, 347–383 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  4. Ando, K., Isozaki, H., Morioka, H.: Spectral properties of Schrödinger operators on perturbed lattices. Ann. Henri Poincaré 17, 2103–2171 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  5. Burago, D., Ivanov, S., Kurylev, Y.: A graph discretization of the Laplace–Beltrami operator. J. Spectr. Theory 4, 675–714 (2014)

    Article  MathSciNet  Google Scholar 

  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)

  7. Chadan, K., Colton, D., Päivärinta, L., Rundell, W.: An Introduction to Inverse Scattering and Inverse Boundary Value Problems. SIAM, Philadelphia (1997)

    Book  Google Scholar 

  8. Chung, R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  9. Colin de Verdière, Y.: Réseaux électriques planaires I. Comment. Math. Helv. 69, 351–374 (1994)

    Article  MathSciNet  Google Scholar 

  10. Colin de Verdière, Y.: Cours Spécialisés 4, Spectre de Graphes. Soc. Math. de France (1998)

  11. Colin de Verdière, Y., Françoise, T.: Scattering theory for graphs isomorphic to a regular tree at infinity. J. Math. Phys. 54, 063502 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  12. Colin de Verdière, Y., de Gitler, I., Vertigan, D.: Réseauxélectriques planaires II. Comment. Math. Helv. 71, 144–167 (1996)

    Article  MathSciNet  Google Scholar 

  13. Cuenin, J.C., Siedentop, H.: Dipoles in graphene have infinitely many bound states. J. Math. Phys. 55, 122304 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  14. Curtis, E.B., Morrow, J.A.: Determining the resistors in a network. SIAM J. Appl. Math. 50, 918–930 (1990)

    Article  MathSciNet  Google Scholar 

  15. Curtis, E.B., Morrow, J.A.: The Dirichlet to Neumann map for a resistor network. SIAM J. Appl. Math. 51, 1011–1029 (1991)

    Article  MathSciNet  Google Scholar 

  16. Curtis, E.B., Morrow, J.A.: Inverse Problems for Electrical Networks. World Scientific, Singapore (2000)

    Book  Google Scholar 

  17. Curtis, E.B., Mooers, E., Morrow, J.A.: Finding the conductors in circular networks. Math. Model. Numer. Anal. 28, 781–813 (1994)

    Article  Google Scholar 

  18. Curtis, E.B., Ingerman, D., Morrow, J.A.: Circular planar graphs and resistor networks. Linear Algbr. Its Appl. 283, 115–150 (1998)

    Article  MathSciNet  Google Scholar 

  19. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284, 787–794 (1984)

    Article  MathSciNet  Google Scholar 

  20. Eskina, M.S.: The direct and the inverse scattering problem for a partial difference equation. Sov. Math. Dokl. 7, 193–197 (1966)

    MathSciNet  Google Scholar 

  21. Faddeev, L.D.: Uniqueness of the inverse scattering problem. Vestnik Leningrad Univ. 11, 126–130 (1956)

    MathSciNet  Google Scholar 

  22. Faddeev, L.D.: Increasing solutions of the Schrödinger equations. Sov. Phys. Dokl. 10, 1033–1035 (1966)

    ADS  Google Scholar 

  23. Faddeev, L.D.: Inverse problem of quantum scattering theory. J. Sov. Math. 5, 334–396 (1976)

    Article  Google Scholar 

  24. Gérard, C., Nier, F.: The Mourre theory for analytically fibred operators. J. Funct. Anal. 152, 202–219 (1989)

    Article  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  26. Higuchi, Y., Shirai, T.: Some spectral and geometric properties for infinite graphs. Contemp. Math. 347, 29–56 (2004)

    Article  MathSciNet  Google Scholar 

  27. Higuchi, Y., Nomura, Y.: Spectral structure of the Laplacian on a covering graph. Euro. J. Comb. 30, 570–585 (2009)

    Article  MathSciNet  Google Scholar 

  28. Hiroshima, F., Sakai, I., Shirai, T., Suzuki, A.: Note on the spectrum of discrete Schrödinger operators. J. Math. Ind. 4, 105–108 (2012)

    MATH  Google Scholar 

  29. Hörmander, L.: The Analysis of Linear Partial Differential Operators III, Pseudodifferential Operators. Springer, Berlin (1985)

    MATH  Google Scholar 

  30. Ikehata, M.: Reconstruction of obstacles from boundary measurements. Wave Motion 30, 205–223 (1999)

    Article  MathSciNet  Google Scholar 

  31. Ikehata, M.: Inverse scattering problems and the enclosure method. Inverse Probl. 20, 533–551 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  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. Isozaki, H., Korotyaev, E.: Inverse problems, trace formulae for discrete Schrödinger operators. Ann. Henri Poincaré 13, 751–788 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  34. Isozaki, H., Morioka, H.: A Rellich type theorem for discrete Schrödinger operators. Inverse Probl. Imaging 8, 475–489 (2014)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  36. Kuroda, S.T.: Scattering theory for differential operators, I, II. J. Math. Soc. Jpn. 25(75–104), 222–234 (1973)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  38. Kobayashi, T., Ono, K., Sunada, T.: Periodic Schrödinger operators on a manifold. Forum Math. 1, 69–79 (1989)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  40. Korotyaev, E., Kutsenko, A.: Zigzag nanoribbons in external electric fields. Asymptot. Anal. 66, 187–206 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Korotyaev, E., Saburova, N.: Schrödinger operators on periodic graphs. J. Math. Anal. Appl. 420, 576–611 (2014)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  43. Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275, 805–826 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  44. Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21, 209–234 (1989)

    Article  MathSciNet  Google Scholar 

  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. Nachman, A.: Reconstruction from boundary measurements. Ann. Math. 128, 531–576 (1988)

    Article  MathSciNet  Google Scholar 

  47. Nachman, A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. 143, 71–96 (1996)

    Article  MathSciNet  Google Scholar 

  48. Nakamura, S.: Modified wave operators for discrete Schrödinger operators with long-range perturbations. J. Math. Phys. 55, 112101 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  51. Oberlin, R.: Discrete inverse problems for Schrödinger and resistor networks. Research archive of Research Experiences for Undergraduates program at University of Washington (2000)

  52. Pestov, L., Uhlmann, G.: Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. Math. 161, 1093–1110 (2005)

    Article  MathSciNet  Google Scholar 

  53. Shaban, W., Vainberg, B.: Radiation conditions for the difference Schrödinger operators. Appl. Anal. 80, 525–556 (2001)

    Article  MathSciNet  Google Scholar 

  54. Shipman, S.P.: Eigenfunctions of unbounded support for embedded eigenvalues of locally perturbed periodic graph operators. Commun. Math. Phys. 332, 605–626 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  55. Shirai, T.: The spectrum of infinite regular line graphs. Trans. Am. Math. Soc. 352, 115–132 (1999)

    Article  MathSciNet  Google Scholar 

  56. Sunada, T.: A periodic Schrödinger operator on abelian cover. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 575–583 (1990)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  58. Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987)

    Article  MathSciNet  Google Scholar 

  59. Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Probl. 25, 123011 (2009)

    Article  ADS  Google Scholar 

  60. Yafaev, D.: Mathematical Scattering Theory. American Mathematical Society, Providence (2009)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering and Computer Science, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime, 790-8577, Japan

    Kazunori Ando

  2. Professor Emeritus, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan

    Hiroshi Isozaki

  3. Faculty of Science and Engineering, Doshisha University, Tataramiyakodani 1-3, Kyotanabe, Kyoto, 610-0394, Japan

    Hisashi Morioka

Authors
  1. Kazunori Ando
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  2. Hiroshi Isozaki
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  3. Hisashi Morioka
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Corresponding author

Correspondence to Hiroshi Isozaki.

Additional information

Communicated by Jan Dereziński.

This work is supported by Grant-in-Aid for Scientific Research (S) 15H05740, (B) 16H0394, (C) 17K05303 and Grant-in-Aid for Young Scientists (B) 16K17630, Japan Society for the Promotion of Science.

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Ando, K., Isozaki, H. & Morioka, H. Inverse Scattering for Schrödinger Operators on Perturbed Lattices. Ann. Henri Poincaré 19, 3397–3455 (2018). https://doi.org/10.1007/s00023-018-0721-3

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  • DOI: https://doi.org/10.1007/s00023-018-0721-3

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