Direct and inverse problems in the model of quantum graphs
Abstract
Here we present a detailed statement about solving of spectral and transport problems for quantum graphs. The practical prescriptions for finding the energy of pi-electrons and current-voltage characteristics for aromatic at the disposal of models of quantum graphs are suggested. We propose an algorithm increasing the precision of numerical solution and discuss the facilities for parallel computations. Here we also suggest the method of inverse problem solution which plays an important role in engineering.
Keywords
Supporting agencies: Sections 1, 5, 8, and 10 were completed under a government contract No. 07.514.11.4146 as part of the Federal Target Program "Research and Development in Priority Areas of Russia's Scientific and Technological Complex Development for 2007-2013" Research and development work on the lot code "2012-1.4-07-514- 0017" "Supercomputer Modeling of Processes for Creating Materials with Specified Optoelectronic Properties for Innovative Technologies" on the topic: "High-performance software package for modeling the electronic and electromechanical properties of nanocarbon objects" (application code "2012-1.4-07-514-0017-023"). Sections 2, 3, 4, 6, 7, and 9 were completed under agreement No. 14.B37.21.0371 on the implementation of research work on the topic "Spectral analysis on hybrid manifolds" (application number in the information computerized system "2012-1.2.2-12-000-1001-047") within the framework of the federal target program "Scientific and scientific-pedagogical
About the Authors
I. S. Lobanov
St. Petersburg National Research University of Information
Technologies, Mechanics, and Optics
Russian Federation
Russian Federation
Saint-Petersburg
E. S. Trifanova
St. Petersburg National Research University of Information
Technologies, Mechanics, and Optics
Russian Federation
Russian Federation
Saint-Petersburg
References
1. Alexander S. Superconductivity of networks. A percolation approach to the effects of disorder // Phys. Rev. B. — 1983. — 27. — P. 1541–1557.
2. Platt J. R., Ruedenberg K., et. al. Free-Electron Theory of Conjugated Molecules — Wiley, 1964.
3. Kuchment P. Graph models of wave propagation in thin structures // Waves in Random Media. — 2002. — V. 12, No. 4. — P. R1–R24.
4. Exner P., Post O. Approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds // Commun. Math. Phys., to appear.
5. Ceresole A., Rasetti M. and Zecchina R. Geometry, topology, and physics of non-Abelian lattices // La Rivista del Nuovo Cimento. — 1998. — V. 21, No. 5. — P. 1–56.
6. Павлов Б. С. Модель потенциала нулевого радиуса с внутренней структурой // ТМФ — 1984. — Т. 59, № 3. — С. 345–353.
7. Staszewska G., Staszewski P., et. al. Many-body tight-binding model for aluminum nanoparticles // Phys. Rev. B. — 2005. — V. 71. — P. 045423.
8. Брюнинг Й., Гейлер В. А., Лобанов И. С. Спектральные свойства операторов Шрёдингера на декорированных графах // Матем. заметки. — 2005. — Т. 77, № 6. — С. 932–935
9. Лобанов И.С., Попов И.Ю. Рассеяние на стыке нанотрубок «зигзаг» и «кресло» // Наносистемы: физика, химия, математика. — 2012. — Т. 3, № 2. — С. 4–26.
10. Golubok A.O., Popov I.U., Mukhin I.S., Lobanov I.S. Creation and study of 2D and 3D carbon nanographs // Physica E. — 2012. — V. 44. — P. 976–980.
11. Korotyaev E., Lobanov I. Schr¨odinger operators on zigzag graphs // Annales Henri Poincare. — 2007. — V. 8, No. 6. — P. 1151–1176.
12. Gharekhanlou B., Khorasani S. Current-Voltage Characteristics of Graphane p-n Junctions // IEEE Transactions on Electron Devices. — 2010. — V. 57, issue 1. — P. 209–214.
13. Savini G., Ferrari A. C., Giustino F. Doped graphane: a prototype high-Tc electron-phonon superconductor // Phys Rev Lett. — 2010. — V. 105. — P. 037002.
14. Cantu-Paz E. A Survey of Parallel Genetic Algorithms // Calculateurs paralleles. — 1998. — V. 10. — P. 141– 171.
15. Graphene — Synthesis, Characterization, Properties and Applications. Ed. by Jian Ru Gong. — InTech, 2011.
16. Graphene: Properties, Synthesis and Applications. Ed. by Zhiping Xu — Nova Science Pub Inc, 2012.
17. Coulter S. A Maple Application for Testing Self-Adjointness on Quantum Graphs // SIAM Undergraduate Research Online (SIURO). — 2012, — V. 5.
18. Лобанов И.С. Программа расчета спектральных и транспортных свойств квантовых графов Qgraph // Свидетельство о регистрации № 2011617733. — 2011.
19. Post O. Spectral Analysis on Graph-Like Spaces. — Springer, 2010.
20. Band R., Sawicki A., Smilansky U. Scattering from isospectral quantum graphs // Journal of Physics A: Mathematical and Theoretical. — 2010. — V. 43, Issue 41. — P. 415201.
21. Smilansky U. Discrete graphs — a paradigm model for quantum chaos // Seminaire Poincare XIV. — 2010. — P. 89–114.
22. Mufthas M.R.M., Rupasinghe C.S. 3D Modelling of Carbon Allotropes Used in Nanotechnology // Fourth Asia International Conference on Mathematical/Analytical Modelling and Computer Simulation (AMS). — 2010. — P. 476–481.
23. Analysis on Graphs and its Applications, Ed by. Exner P. et al. — Proc. Symp. Pure Math., AMS, 2008.
24. Rafii-Tabar H. Computational Physics of Carbon Nanotubes. — Cambridge University Press, Cambridge, New York, 2007. — 507 p.
25. Korotyaev E., Lobanov I. Zigzag periodic nanotube in magnetic field // Preprint arXiv:math/0604007.
26. Fan Chung and Linyuan Lu. Complex graphs and networks // CBMS Regional Conference Series in Mathematics. — 2006. — No. 107.
27. Quantum Graphs and Their Applications, Ed. By Berkolaiko G. et al. // Contemp. Math. — 2006. — V. 415.
28. Albeverio S., Gesztesy F., Hoegh-Krohn R., and Holden H. Solvable Models in Quantum Mechanics: Second Edition // AMS Chelsea Publishing. — 2005. — V. 350.
29. Покорный Ю.В., Пенкин О.М., и др. Дифференциальные уравнения на геометрических графах — М.:ФИЗМАТЛИТ, 2004.
30. Bruning J., Geyler V. A. Scattering on compact manifolds with infinitely thin horns // J. Math. Phys. — 2003. — V. 44. — P. 371–405.
31. Liu Y. J., Chen X. L. Continuum Models of Carbon Nanotube-Based Composites Using the Boundary Element Method // Electronic Journal of Boundary Elements. — 2003. — V. 1, No. 2. — P. 316–335.
32. Dequesnes M., Rotkin S. V., Aluru N. R. Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches // Nanotechnology. — 2002. — V. 13. — P. 120–131.
33. Van Lier G., Van Alsenoy C., Van Doren V., Geerlings P. Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene // Chemical Physics Letters. — 2000. — V. 326, Issues 1–2. — P. 181–185.
34. Marx D., Hutter J. Ab-initio Molecular Dynamics: Theory and Implementation // Modern Methods and Algorithms of Quantum Chemistry, Proceedings. — 2000. — V. 3. — P. 329–477.
35. Ceresole A., Rasetti M., Zecchina R. Geometry, topology, and physics of non-Abelian lattices // La Rivista del Nuovo Cimento. — 1998. — V. 21, No. 5. — P. 1–56.
36. Fan Chung. Spectral Graph Theory // CBMS Regional Conference Series in Mathematics. — 1997. — No. 92.
37. Avron J. E., Raveh A., Zur B. Adiabatic quantum transport in multiply connected systems // Rev. Modern Phys. — 1988. — V. 60, No. 4. — P. 873–915.
38. Ruedenberg K., Scherr C.W. Free-electron network model for conjugated systems. I. Theory // J. Chem. Phys. — 1953. — V. 21. — P. 1565–1581.
39. Kuhn H. Elektronengasmodell zur quantitativen Deutung der Lichtabsorption von organischen Farbstoffen // Helv. Chim. Acta. — 1948. — V. 31. — P. 1441.
40. Pauling L. The Diamagnetic Anisotropy of Aromatic Molecules // J. Chem. Phys. — 1936. — V. 4. — P. 673.
Review
For citations:
Lobanov I.S., Trifanova E.S. Direct and inverse problems in the model of quantum graphs. Nanosystems: Physics, Chemistry, Mathematics. 2012;3(5):6-32. (In Russ.)
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