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Differential and Pseudo-differential Operators on Graphs as Models of Mesoscopic Systems

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Analysis and Applications — ISAAC 2001

Part of the book series: International Society for Analysis, Applications and Computation ((ISAA,volume 10))

Abstract

The lecture contains a brief survey on graph models for wave propagation in thin structures.

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References

  1. Alexander, S.: Superconductivity of networks. A percolation approach to the effects of disorder. Phys. Rev. B, 27 (1983), 1541–1557.

    Article  MathSciNet  Google Scholar 

  2. Ashcroft, N.W., Mermin, N.D.: (1976) Solid State Physics. New York-London: Holt, Rinehart and Winston.

    Google Scholar 

  3. Aviram, A., Ratner, M. (Eds.): (1998). Molecular Electronics: Science and Technology. Ann. New York Acad. Sci., Vol. 852.

    Google Scholar 

  4. Avishai, Y., Bessis, D., Giraud, B.G., Mantica, G.: Quantum bound states in open geometries. Phys. Rev. B 44 (1991), no. 15, 8028–8034.

    Article  Google Scholar 

  5. Avishai, Y., Luck, J.M.: Quantum percolation and ballistic conductance on a lattice of wires. Phys. Rew. B 45 (1992), no. 3, 1074–1095.

    Article  Google Scholar 

  6. Avron, J., Exner, P., Last, Y.: Periodic Schrödinger operators with large gaps and Wannier-Stark ladders. Phys. Rev. Lett. 72 (1994), 869–899.

    Article  Google Scholar 

  7. Avron, J., Raveh, A., Zur, B.: Adiabatic quantum transport in multiply connected systems. Rev. Mod. Phys. 60 (1988), no. 4, 873–915.

    Article  MathSciNet  Google Scholar 

  8. Axmann, W.,Kuchment, P., Kunyansky, L.: Asymptotic methods for thin high contrast 2D PBG materials. Journal of Lightwave Technology, 17 (1999), no. 11, 1996–2007.

    Google Scholar 

  9. Benchama, N., Kuchment, P.: Asymptotics models for acoustic waves in high contrast media, in preparation.

    Google Scholar 

  10. Bensoussan, A., Lions, J.L., Papanicolaou, G. (1980): Asymptotic Analysis of Periodic Structures. Amsterdam: North-Holland.

    Google Scholar 

  11. Birman, M.Sh., Suslina, T.A.: Periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity. Algebra u Analiz 11(1999), no.2. English translation in St. Petersburg Math J. 11 (2000), no. 2, 203–232.

    MathSciNet  Google Scholar 

  12. Birman, M.Sh., Suslina, T.A., Shterenberg, R.G.: Absolute continuity of the spectrum of a two-dimensional Schrödinger operator with potential supported on a periodic system of curves. Preprint ESI no. 934, http://www.esi.ac.at, 2000, to appear in Algebra i Analiz (St. Petersburg Math. J.).

    Google Scholar 

  13. Boroditsky, M., Vrijen, R., Krauss, T.F., Coccioli, R., Bhat, R., Yablonovitch, E.: Spontaneous emission extraction and Purcell enhancement from thin-film 2D photonic crystals. Journal of Lightwave Technology, 17 (1999), no. 11, 2096–2112.

    Article  Google Scholar 

  14. Carini, J.P., Londergan, J.T., Murdock, D.P., Trinkle, D., Yung, C.S.: Bound states in waveguides and bent quantum wires. I. Applications to waveguide systems. Phys. Rev. B 55 (1997), 9842–9851.

    Google Scholar 

  15. Carini, J.P., Londergan, J.T., Murdock, D.P.: Bound states in waveguides and bent quantum wires. II. Electrons in quantum wires. Phys. Rev. B 55 (1997), 9852–9859.

    Article  Google Scholar 

  16. Carlson, R.: Hill’s equation for a homogeneous tree. Electronic J. Duff. Equations (1997), no. 23, 1–30

    Google Scholar 

  17. Carlson, R.: Adjoint and self-adjoint operators on graphs. Electronic J. Duff. Equations (1998), no. 6, 1–10

    Google Scholar 

  18. Carlson, R.: Inverse eigenvalue problems on directed graphs. Trans. Amer. Math. Soc. 351 (1999), no. 10, 4069–4088.

    Article  MathSciNet  MATH  Google Scholar 

  19. Carlson, R.: Nonclassical Sturm-Liouville problems and Schrödinger Operators on Radial Trees. Preprint 2000.

    Google Scholar 

  20. de Gennes, P.-G.: Champ critique d’une boucle supraconductrice ramefiee. C. R. Acad. Sc. Paris 292B (1981), 279–282.

    Google Scholar 

  21. Datta, S. (1999): Electronic Transport in Mesoscopic Systems. Cambridge: Cambridge Univ. Press.

    Google Scholar 

  22. Evans, W.D., Harris, D.J.: Fractals, trees and the Neumann Laplacian. Math. Ann. 296 (1993), 493–527.

    Article  MathSciNet  MATH  Google Scholar 

  23. Evans, W.D., Saito, Y.: Neumann Laplacians on domains and operators on associated trees, to appear in Quart. J. Math. Oxford.

    Google Scholar 

  24. Exner, P., Seba, P.: Electrons in semiconductor microstructures: a challenge to operator theorists. Schrödinger Operators, Standard and Nonstandard (Dubna 1988). Singapore: World Scientific, Singapore 1989; pp. 79–100.

    Google Scholar 

  25. Figotin, A.: High-contrast photonic crystals. Diffuse waves in complex media (Les Houches, 1998 ), 109–136, NATO Sci. Ser. C Math. Phys. Sci., 531. Dordrecht: Kluwer Acad. Publ. 1999.

    Google Scholar 

  26. Figotin, A., Godin, Yu.: Spectral properties of thin-film photonic crystals. SIAM J. Appl. Math., 61 (2001), no. 6, 1959–1979.

    MathSciNet  MATH  Google Scholar 

  27. Figotin, A., Kuchment, P.: Band-Gap Structure of the Spectrum of Periodic and Acoustic Media. I. Scalar Model. SIAM J. Applied Math. 56 (1996), no. 1, 68–88.

    MathSciNet  MATH  Google Scholar 

  28. Figotin, A., Kuchment, P.: Band-Gap Structure of the Spectrum of Periodic and Acoustic Media. II. 2D Photonic Crystals. SIAM J. Applied Math. 56 (1996), 1561–1620.

    MathSciNet  MATH  Google Scholar 

  29. Figotin, A., Kuchment, P.: Spectral properties of classical waves in high contrast periodic media. SIAM J. Appl. Math. 58 (1998), no. 2, 683–702.

    MathSciNet  MATH  Google Scholar 

  30. Figotin, A., Kuchment, P.: Asymptotic models of high contrast periodic photonic and acoustic media (tentative title). Parts I and II, in preparation.

    Google Scholar 

  31. Freidlin, M. (1996): Markov Processes and Differential Equations: Asymptotic Problems, Lectures in Mathematics ETH Zürich. Basel: Birkhäuser Verlag.

    Google Scholar 

  32. Freidlin, M., Wentzell, A.: Diffusion processes on graphs and the averaging principle. Annals of Probability, 21 (1993), no. 4, 2215–2245.

    Article  MathSciNet  MATH  Google Scholar 

  33. Friedlander, L.: On the density of states in periodic media in the large coupling limit. To appear in Comm Partial Diff. Equations.

    Google Scholar 

  34. Gerasimenko, N., Pavlov, B.: Scattering problems on non-compact graphs. The-or. Math. Phys., 75 (1988), 230–240.

    MathSciNet  Google Scholar 

  35. Giovannella, C., Lambert, C.J. (Editors) (1998): Lectures on Superconductivity in Networks and Mesoscopic Systems: Pontignano, Italy September 1997 (Aip Conference Proceedings, Vol 427). Amer Inst of Physics.

    Google Scholar 

  36. Griffith, J.S.: A free-electron theory of conjugated molecules. I. Polycyclic Hydrocarbons. Trans. Faraday Soc., 49 (1953), 345–351.

    Article  Google Scholar 

  37. Griffith, J.S.: A free-electron theory of conjugated molecules. II. A derived algebraic scheme. Proc. Camb. Philos. Soc., 49 (1953), 650–658.

    Article  MATH  Google Scholar 

  38. Hempel, R., Lienau, K.: Spectral properties of periodic media in the large coupling limit. Comm Partial Diff. Equations 25 (2000), 1445–1470.

    MathSciNet  MATH  Google Scholar 

  39. Imry, Y. (1997): Introduction to Mesoscopic Physics. Oxford: OxfordUniversity Press.

    Google Scholar 

  40. Inoue, K., Sasada, M., Kuwamata, J., Sakoda, K., Haus, J.W.: A two-dimensional photonic crystal laser. Japan J. Appl. Phys. 85(1999), no. 8, 57685770.

    Google Scholar 

  41. Jackson, J.D. (1962): Classical Electrodynamics. New York: John Wiley & Sons.

    Google Scholar 

  42. Joachim, C., Roth, S. (Eds.) (1997): Atomic and Molecular Wires. NATO Adv. Ser.: E Appl. Sci., Vol. 341. Dordrecht: Kluwer.

    Google Scholar 

  43. Joannopoulos, J.D., Meade, R.D., Winn, J.N. (1995): Photonic Crystals, Molding the Flow of Light. Princeton: Princeton Univ. Press.

    Google Scholar 

  44. John, S.: Strong localization of photons in certain disordered dielectric super-lattices. Phys. Rev. Lett. 58 (1987), 2486.

    Google Scholar 

  45. Kostrykin, V., Schrader, R.: Kirchgoff’s rule for quantum wires. J. Phys. A 32 (1999), 595–630.

    Article  MathSciNet  MATH  Google Scholar 

  46. Kostrykin, V., Schrader, R.: Kirchgoff’s rule for quantum wires. II: The inverse problem with possible applications to quantum computers. Preprint 2000.

    Google Scholar 

  47. Kottos, T., Smilansky, U.: Quantum chaos on graphs. Phys. Rev. Lett. 79 (1997), 4794–4797.

    Article  Google Scholar 

  48. Kuchment, P. (1993): Floquet Theory for Partial Differential Equations. Basel: Birkhäuser Verlag.

    Book  MATH  Google Scholar 

  49. Kuchment, P.: The Mathematics of Photonics Crystals. Ch. 7 in Mathematical Modeling in Optical Science. Bao, G., Cowsar, L. and Masters, W. (Editors). (2001). Philadelphia: SIAM.

    Google Scholar 

  50. Kuchment, P.: Opening spectral gaps for differential operators on graphs using Schenker-Aizenman decoration procedure. In preparation

    Google Scholar 

  51. Kuchment, P., Kunyansky, L.: Spectral Properties of High Contrast Band-Gap Materials and Operators on Graphs. Experimental Mathematics, 8 (1999), no. 1, 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  52. Kuchment, P., Kunyansky, L.: Differential operators on graphs and photonic crystals. To appear in Adv. Comput. Math.

    Google Scholar 

  53. Kuchment, P., Zeng, H.: Convergence of spectra of mesoscopic systems collapsing onto a graph. J. Math. Anal. Appl. 258 (2001), 671–700.

    Article  MathSciNet  MATH  Google Scholar 

  54. Kuchment, P., Zeng, H.: Convergence of spectra of mesoscopic systems collapsing onto a graph II. Large protrusions at vertices. In preparation.

    Google Scholar 

  55. Melnikov, Yu.B., Pavlov, B.S.: Two-body scattering on a graph and application to simple nanoelectronic devices. J. Math. Phys. 36 (1995), 2813–2825.

    Google Scholar 

  56. Mittra, R., Lee, S.W. (1971): Analytic Techniques in the Theory of Guided Waves. London: Collier - Macmillan.

    Google Scholar 

  57. Murayama, Y. (2001): Mesoscopic Systems. John Wiley & Sons.:

    Google Scholar 

  58. Naimark, K., Solomyak, M.: Eigenvalue estimates for the weighted Laplacian on metric trees. Proc. London Math. Soc., 80 (2000), 690–724.

    MathSciNet  MATH  Google Scholar 

  59. Novikov, S.: Schrödinger operators on graphs and symplectic geometry. The Arnoldfest (Toronto, ON, 1997 ), 397–413, Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  60. Pavlov, B.S.: A model of zero-radius potential with internal structure. Theor. Math. Phys. 59 (1984), 544–580.

    Article  Google Scholar 

  61. Pavlov, B.S.: The theory of extensions and explicitly solvable models. Russian. Math. Surveys 42 (1987), 127–168.

    Article  MATH  Google Scholar 

  62. Photonic & Acoustic Band-Gap Bibliography, http://home.earthlink.net/ jpdowling/pbgbib.html

    Google Scholar 

  63. Reed, M., Simon, B. (1978): Methods of Modern Mathematical Physics, I V: Analysis of Operators. New York: Academic Press.

    MATH  Google Scholar 

  64. Rubinstein, J., Schatzman, M.: Spectral and variational problems on multiconnected strips. C. R. Acad. Sci. Paris Ser. I math. 325 (1997), no. 4, 377–382.

    Article  MathSciNet  MATH  Google Scholar 

  65. Rubinstein, J., Schatzman, M.: Asymptotics for thin superconducting rings. J. Math. Pures Appl. (9) 77 (1998), no. 8, 801–820.

    MathSciNet  Google Scholar 

  66. Rubinstein, J., Schatzman, M.: On multiply connected mesoscopic superconducting structures. Semin. Theor. Spectr. Geom., no. 15, Univ. Grenoble I, Saint-Martin-d’Heres, 1998, 207–220.

    Google Scholar 

  67. Rubinstein, J., Schatzman, M.: Variational problems on multiply connected thin strips I: Basic estimates and convergence of the Laplacian spectrum. Preprint 1999.

    Google Scholar 

  68. Ruedenberg, K., Scherr, C.W.: Free-electron network model for conjugated systems. I. Theory. J. Chem. Physics, 21 (1953), no. 9, 1565–1581.

    Article  Google Scholar 

  69. Saito, Y.: The limiting equation of the Neumann Laplacians on shrinking domains. Preprint 1999.

    Google Scholar 

  70. Sakoda, K. (2001): Optical Properties of Photonic Crystals. Berlin: Springer Verlag.

    Google Scholar 

  71. Schatzman, M.: On the eigenvalues of the Laplace operator on a thin set with Neumann boundary conditions. Applicable Anal. 61 (1996), 293–306.

    Article  MathSciNet  Google Scholar 

  72. Schenker, J., Aizenman, M.: The creation of spectral gaps by graph decoration. Lett. Math. Phys. 53 (2000), no. 3, 253.

    Article  MathSciNet  MATH  Google Scholar 

  73. Serena, P.A., Garcia, N. (Eds.) (1997): Nanowires. NATO Adv. Ser.: E Appl. Sci., Vol. 340, Dordrecht: Kluwer.

    Google Scholar 

  74. Shepherd, T.J., Roberts, P.J.: Soluble two-dimensional photonic-crystal model. Phys. rev. E 55 (1997), no. 5, 6024–6038.

    Article  Google Scholar 

  75. Sievenpiper, D., Zhang, L., Jimenez Broas, R.F., Alexdpoulos, N.A., Yablonovitch, E.: High-impedance electromagnetic surfaces with a forbidden frequency band. IEEE Trans. Microwave Theory and Tech. 47 (1999), no. 11, 2059–2074.

    Article  Google Scholar 

  76. Thomas, L.E.: Time dependent approach to scattering from impurities in a crystal. Comm. Math. Phys. 33 (1973), 335–343.

    Article  MathSciNet  Google Scholar 

  77. Thornton, T.J.: Mesoscopic devices. Rep. Prog. Phys. 57 (1994), 311–364.

    Google Scholar 

  78. Yablonovitch, E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58: 2059 (1987).

    Article  Google Scholar 

  79. Zeng, H.: Convergence of spectra of mesoscopic systems collapsing onto a graph. PhD Thesis, Wichita State University, Wichita, KS 2001.

    Google Scholar 

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Authors and Affiliations

  1. Texas A & M University, College Station, TX, USA, 77843

    Peter Kuchment

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  1. Peter Kuchment
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Editors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    Heinrich G. W. Begehr

  2. University of Delaware, Newark, DE, USA

    Robert P. Gilbert

  3. York University, Toronto, Canada

    Man Wah Wong

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Kuchment, P. (2003). Differential and Pseudo-differential Operators on Graphs as Models of Mesoscopic Systems. In: Begehr, H.G.W., Gilbert, R.P., Wong, M.W. (eds) Analysis and Applications — ISAAC 2001. International Society for Analysis, Applications and Computation, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3741-7_2

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  • DOI: https://doi.org/10.1007/978-1-4757-3741-7_2

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