Abstract
We consider the spectral theory and inverse scattering problem for discrete Schrödinger operators on the hexagonal lattice. We give a procedure for reconstructing finitely supported potentials from the scattering matrices for all energies. The same procedure is applicable for the inverse scattering problem on the triangle lattice.
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Ablowitz M.J., Nachman A.I.: A multidimensional inverse-scattering method. Stud. Appl. Math. 71(3), 243–250 (1984)
Beals, R., Coifman R.R.: Multidimensional inverse scatterings and nonlinear partial differential equations. In: Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984). Proc. Sympos. Pure Math., vol. 43, pp. 45–70. Amer. Math. Soc., Providence (1985)
Birman M.Š.: Existence conditions for wave operators. Izv. Akad. Nauk SSSR Ser. Mat. 27, 883–906 (1963)
Case K.M., Kac M.: A discrete version of the inverse scattering problem. J. Math. Phys. 14, 594–603 (1973)
Castro Neto A.H., Guinea F., Peres N.M.R., Novoselov K.S., Geim A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81(1), 109–162 (2009)
Chung, F.R.K.: Spectral graph theory. In: CBMS Regional Conference Series in Mathematics, vol. 92. Conference Board of the Mathematical Sciences, Washington (1997)
Faddeev L.D.: Uniqueness of solution of the inverse scattering problem. Vestnik Leningrad. Univ. 11(7), 126–130 (1956)
Faddeev L.D.: Inverse problem of quantum scattering theory. II. J. Math. Sci. 5, 334–396 (1976). doi:10.1007/BF01083780
Gel’fand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Izvestiya Akad. Nauk SSSR. Ser. Mat. 15, 309–360 (1951); Am. Math. Soc. Transl. 2(1), 253–304 (1955)
González J., Guinea F., Vozmediano M.A.H.: The electronic spectrum of fullerenes from the Dirac equation. Nuclear Phys. B 406(3), 771–794 (1993)
Isozaki H.: Inverse scattering theory for Dirac operators. Ann. Inst. H. Poincaré Phys. Théor. 66(2), 237–270 (1997)
Isozaki, H.: Inverse spectral theory. In: Topics in the Theory of Schrödinger Operators, pp. 93–143. World Scientific, River Edge (2004)
Isozaki, H., Korotayev, E.L.: Inverse problems, trace formulae for discrete Schrödinger operators. Annales de l’Institut Henri Poincaré (2012, to appear)
Kato T.: On finite-dimensional perturbations of self-adjoint operators. J. Math. Soc. Japan 9, 239–249 (1957)
Kato T., Kuroda S.T.: The abstract theory of scattering. Rocky Mt. J. Math. 1(1), 127–171 (1971)
Khenkin, G.M., Novikov, R.G.: The \({\overline\partial}\) -equation in the multidimensional inverse scattering problem. Uspekhi Mat. Nauk 42(3), 93–152, 255 (1987)
Korotyaev E.L., Kutsenko A.: Zigzag nanoribbons in external electric fields. Asymptot. Anal. 66(3–4), 187–206 (2010)
Kotani M., Shirai T., Sunada T.: Asymptotic behavior of the transition probability of a random walk on an infinite graph. J. Funct. Anal. 159(2), 664–689 (1998)
Kuchment P., Post O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275(3), 805–826 (2007)
Kuroda S.T.: Perturbation of continuous spectra by unbounded operators. I. J. Math. Soc. Japan 11, 246–262 (1959)
Kuroda S.T.: Perturbation of continuous spectra by unbounded operators. II. J. Math. Soc. Japan 12, 243–257 (1960)
Kuroda S.T.: Scattering theory for differential operators. I. Operator theory. J. Math. Soc. Japan 25, 75–104 (1973)
Kuroda S.T.: Scattering theory for differential operators. II. Self-adjoint elliptic operators. J. Math. Soc. Japan 25, 222–234 (1973)
Marčenko V.A.: On reconstruction of the potential energy from phases of the scattered waves. Dokl. Akad. Nauk SSSR (N.S.) 104, 695–698 (1955)
Mourre E.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78(3), 391–408 (1980)
Newton, R.G.: The Gel’fand-Levitan method in the inverse scattering problem. In: Lavita, J.A., Dordrecht, J.P., Marchand, Reidel, D. (eds.) Scattering Theory in Mathematical Physics (1974)
Reed M., Simon B.: Methods of Modern Mathematical Physics. II. Academic Press, New York (1979)
Reed M., Simon B.: Methods of modern mathematical physics. III. Academic Press, New York (1975)
Rosenblum M.: Perturbation of the continuous spectrum and unitary equivalence. Pac. J. Math. 7, 997–1010 (1957)
Semenoff G.W.: Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53(26), 2449–2452 (1984)
Weder R.: Characterization of the scattering data in multidimensional inverse scattering theory. Inverse Problems 7(3), 461–489 (1991)
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Communicated by Jan Derezinski.
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Ando, K. Inverse Scattering Theory for Discrete Schrödinger Operators on the Hexagonal Lattice. Ann. Henri Poincaré 14, 347–383 (2013). https://doi.org/10.1007/s00023-012-0183-y
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DOI: https://doi.org/10.1007/s00023-012-0183-y