Schrödinger operators with guided potentials on periodic graphs

Korotyaev, Evgeny; Saburova, Natalia

doi:10.1090/proc/13733

Schrödinger operators with guided potentials on periodic graphs
HTML articles powered by AMS MathViewer

by Evgeny Korotyaev and Natalia Saburova PDF

Proc. Amer. Math. Soc. 145 (2017), 4869-4883 Request permission

Abstract:

We consider discrete Schrödinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the unperturbed operator is a union of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed operator consists of the “unperturbed” one plus the additional guided spectrum, which is a union of a finite number of bands. We estimate the position of the guided bands and their length in terms of graph geometric parameters. We also determine the asymptotics of the guided bands for large guided potentials. Moreover, we show that the possible number of the guided bands, their length and position can be rather arbitrary for some specific potentials.

References

Kazunori Ando, Inverse scattering theory for discrete Schrödinger operators on the hexagonal lattice, Ann. Henri Poincaré 14 (2013), no. 2, 347–383. MR 3028042, DOI 10.1007/s00023-012-0183-y
Gregory Berkolaiko and Peter Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208, DOI 10.1090/surv/186
Anne Boutet de Monvel and Jaouad Sahbani, On the spectral properties of discrete Schrödinger operators: the multi-dimensional case, Rev. Math. Phys. 11 (1999), no. 9, 1061–1078. MR 1725827, DOI 10.1142/S0129055X99000337
A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81 (2009), 109–162.
D. J. Colquitt, M. J. Nieves, I. S. Jones, A. B. Movchan, and N. V. Movchan, Localization for a line defect in an infinite square lattice, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2013), no. 2150, 20120579, 23. MR 3003822, DOI 10.1098/rspa.2012.0579
D. Gieseker, H. Knörrer, and E. Trubowitz, The geometry of algebraic Fermi curves, Perspectives in Mathematics, vol. 14, Academic Press, Inc., Boston, MA, 1993. MR 1184395
Batu Güneysu, Semiclassical limits of quantum partition functions on infinite graphs, J. Math. Phys. 56 (2015), no. 2, 022102, 13. MR 3390865, DOI 10.1063/1.4907385
R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D.-X. Xu, J. G. Wanguemert-Perez, I. Molina-Fernandez, and S. Janz, Waveguide sub-wavelength structures: a review of principles and applications, Laser Photon. Rev., 9 (2015), 25–49.
P. Harris, Carbon nano-tubes and related structure, Cambridge, Cambridge University Press, 2002.
W. A. Harrison, Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond, Dover Publications, Inc., New York, 1989.
Yusuke Higuchi and Yuji Nomura, Spectral structure of the Laplacian on a covering graph, European J. Combin. 30 (2009), no. 2, 570–585. MR 2489251, DOI 10.1016/j.ejc.2008.03.008
Yusuke Higuchi and Tomoyuki Shirai, The spectrum of magnetic Schrödinger operators on a graph with periodic structure, J. Funct. Anal. 169 (1999), no. 2, 456–480. MR 1730561, DOI 10.1006/jfan.1999.3478
Yusuke Higuchi and Tomoyuki Shirai, A remark on the spectrum of magnetic Laplacian on a graph, Proceedings of the 10th Workshop on Topological Graph Theory (Yokohama, 1998), 1999, pp. 129–141. MR 1732772
Yusuke Higuchi and Tomoyuki Shirai, Some spectral and geometric properties for infinite graphs, Discrete geometric analysis, Contemp. Math., vol. 347, Amer. Math. Soc., Providence, RI, 2004, pp. 29–56. MR 2077029, DOI 10.1090/conm/347/06265
Hiroshi Isozaki and Evgeny Korotyaev, Inverse problems, trace formulae for discrete Schrödinger operators, Ann. Henri Poincaré 13 (2012), no. 4, 751–788. MR 2913620, DOI 10.1007/s00023-011-0141-0
Hiroshi Isozaki and Hisashi Morioka, A Rellich type theorem for discrete Schrödinger operators, Inverse Probl. Imaging 8 (2014), no. 2, 475–489. MR 3209307, DOI 10.3934/ipi.2014.8.475
S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Linear waveguides in photonic crystal slabs, Phys. Rev. B, 62 (2000), 8212–8222.
S. G. Johnson and J. D. Joannopoulos, Photonic crystals. The road from theory to practice, Springer US, 2002.
N. I. Karachalios, The number of bound states for a discrete Schrödinger operator on $\Bbb Z^N,\ N\geq 1$, lattices, J. Phys. A 41 (2008), no. 45, 455201, 14. MR 2515832, DOI 10.1088/1751-8113/41/45/455201
E. L. Korotyaev, Inverse resonance scattering for Jacobi operators, Russ. J. Math. Phys. 18 (2011), no. 4, 427–439. MR 2863561, DOI 10.1134/S1061920811040054
Evgeny L. Korotyaev and Anton Kutsenko, Zigzag nanoribbons in external electric fields, Asymptot. Anal. 66 (2010), no. 3-4, 187–206. MR 2648784
Evgeny L. Korotyaev and Anton A. Kutsenko, Zigzag nanoribbons in external electric and magnetic fields, Int. J. Comput. Sci. Math. 3 (2010), no. 1-2, 168–191. MR 2682280, DOI 10.1504/IJCSM.2010.033933
E. Korotyaev and A. Laptev, Trace formulae for Schrödinger operators with complex-valued potentials on cubic lattices, preprint: arXiv:1609.09703, 2016.
Evgeny Korotyaev and Natalia Saburova, Schrödinger operators on periodic discrete graphs, J. Math. Anal. Appl. 420 (2014), no. 1, 576–611. MR 3229841, DOI 10.1016/j.jmaa.2014.05.088
Evgeny Korotyaev and Natalia Saburova, Spectral band localization for Schrödinger operators on discrete periodic graphs, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3951–3967. MR 3359585, DOI 10.1090/S0002-9939-2015-12586-5
Evgeny Korotyaev and Natalia Saburova, Effective masses for Laplacians on periodic graphs, J. Math. Anal. Appl. 436 (2016), no. 1, 104–130. MR 3440084, DOI 10.1016/j.jmaa.2015.11.051
Evgeny Korotyaev and Natalia Saburova, Magnetic Schrödinger operators on periodic discrete graphs, J. Funct. Anal. 272 (2017), no. 4, 1625–1660. MR 3590247, DOI 10.1016/j.jfa.2016.12.015
Anton A. Kutsenko, Wave propagation through periodic lattice with defects, Comput. Mech. 54 (2014), no. 6, 1559–1568. MR 3275237, DOI 10.1007/s00466-014-1076-3
Anton A. Kutsenko, Algebra of 2D periodic operators with local and perpendicular defects, J. Math. Anal. Appl. 442 (2016), no. 2, 796–803. MR 3504027, DOI 10.1016/j.jmaa.2016.05.015
Fernando Lledó and Olaf Post, Eigenvalue bracketing for discrete and metric graphs, J. Math. Anal. Appl. 348 (2008), no. 2, 806–833. MR 2446037, DOI 10.1016/j.jmaa.2008.07.029
K. S. Novoselov and A. K. Geim, et al., Electric field effect in atomically thin carbon films, Science 22 October, 306 (2004), no. 5696, 666–669.
G. G. Osharovich and M. V. Ayzenberg-Stepanenko, Wave localization in stratified square-cell lattices: The antiplane problem, J. Sound Vib., 331 (2012), 1378–1397.
D. Parra and S. Richard, Spectral and scattering theory for Schrödinger operators on perturbed topological crystals, preprint: arXiv:1607.03573, 2016.
Vladimir S. Rabinovich and Steffen Roch, Essential spectra of difference operators on $\Bbb Z^n$-periodic graphs, J. Phys. A 40 (2007), no. 33, 10109–10128. MR 2371282, DOI 10.1088/1751-8113/40/33/012
Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
Grigori Rozenblum and Michael Solomyak, On the spectral estimates for the Schrödinger operator on $\Bbb Z^d,\ d\ge 3$, J. Math. Sci. (N.Y.) 159 (2009), no. 2, 241–263. Problems in mathematical analysis. No. 41. MR 2544038, DOI 10.1007/s10958-009-9436-9
Tomoyuki Shirai, A trace formula for discrete Schrödinger operators, Publ. Res. Inst. Math. Sci. 34 (1998), no. 1, 27–41. MR 1617621, DOI 10.2977/prims/1195144826
Morikazu Toda, Theory of nonlinear lattices, 2nd ed., Springer Series in Solid-State Sciences, vol. 20, Springer-Verlag, Berlin, 1989. MR 971987, DOI 10.1007/978-3-642-83219-2
Esa V. Vesalainen, Rellich type theorems for unbounded domains, Inverse Probl. Imaging 8 (2014), no. 3, 865–883. MR 3295949, DOI 10.3934/ipi.2014.8.865