Abstract
The quantum description of transport properties in nanostructures are directly connected with the geometry of the object and the corresponding electron states in a field of atomic systems [1, 2]. Waveguide properties in optics and microwaves demonstrate the very rich set of possibilities for solitonic behavior [3], and also constructive possibilities [4]. The conventional quantum mechanics of pure electron states originated from the mathematical results of Floquet [5], which lead to the fundamental notion of the Bloch function. The quantum state is here defined as the common eigenfunction of commuting translational symmetric Hamiltonian and shift operators [6]
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References
D. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003)
A. Blanco-Redondo, I. Andonegui, M.J. Collins, G. Harari, Y. Lumer, M.C. Rechtsman, B.J. Eggleton, M. Segev, Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States. Phys. Rev. Lett. 116, 163901 (2016)
A.A. Sukhorukov, Y.S. Kivshar, H.S. Eisenberg, Y. Silberberg, Spatial optical solitons in waveguide arrays. IEEE J. Quantum Electron. 39, 31–50 (2003)
R.S. Savelev, A.P. Slobozhanyuk, A.E. Miroshnichenko, Y.S. Kivshar, P.A. Belov, Subwavelength waveguides composed of dielectric nanoparticles. Phys. Rev. B 89, 035435 (2014)
G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques. Ann. de l’Ecole Norm. Supérieure 12, 47–88 (1883)
V.A. Geyler, IYu. Popov, Group-theoretical analysis of lattice Hamiltonians with a magnetic field. Phys. Lett. A 201, 359–364 (1995)
S.B. Leble, Kolmogorov equation for Bloch electrons and electrical resistivity models for nanowires. Nanosyst. Phys. Chem. Math. 8(2), 247–259 (2017)
E. Fermi, Sopra lo spostamento per pressione delle righe elevate delle serie spettrali. Il Nuovo Cim. 11, 157–166 (1934)
G. Breit, The scattering of slow neutrons by bound protons: methods of calculation. Phys. Rev. 71, 215 (1947)
YuN Demkov, V.N. Ostrovskii, Zero-Range Potentials and Their Applications in Atomic Physics (Plenum, New York, 1988)
B.S. Pavlov, The theory of extensions and explicitly-solvable models. Uspekhi Mat. Nauk. 258, 99–131 (1987); Russ. Math. Surv. 42 (6), 127–168 (1987)
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics (Springer, New York, 1988)
S.B. Leble, S. Yalunin, Multiple scattering and electron-uracil collisions at low energies. EPJ 144, 115–122 (2007)
K. Huang, C.N. Yang, Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev. 105, 767–775 (1957)
A. Derevianko, Revised Huang-Yang multipolar pseudopotential. Phys. Rev. A 72, 044701 (2005)
S.B. Leble, S. Yalunin, A dressing of zero-range potentials and electron-molecule scattering problem at low energies. Phys. Lett. A 339, 83–88 (2005)
S.B. Leble, S. Yalunin, Generalized zero range potentials and multi-channel electron-molecule scattering. Rad. Phys. Chem. 68, 181–186 (2003)
S. Leble, D.V. Ponomarev, Dressing of zero-range potentials into realistic molecular potentials of finite range. Task Q. 14(1–2), 29–34 (2010)
V.M. Adamyan, I.V. Blinova, A.I. Popov, I.Yu. Popov: Waveguide bands for a system of macromolecules. Nanosyst. Phys. Chem. Math. 6(5), 611–617 (2015)
E.N. Grishanov, I.Y. Popov, Electron spectrum for aligned SWNT array in a magnetic field. Superlattices Microstruct. 100, 1276–1282 (2016)
E.N. Grishanov, I.Y. Popov, Computer simulation of periodic nanostructures. Nanosyst. Phys. Chem. Math. 7(5), 865–868 (2016)
S. Leble, Cyclic periodic ZRP structures. Scattering problem for generalized Bloch functions and conductivity. Nanosyst. Phys. Chem. Math. 9 (2), 225–243 (2018)
D.V. Ponomarev, Electronic states in zero-range potential models of nanostructures with a cyclic symmetry. M.Sc. Thesis. Supervisor: S. Leble, Gdansk University of Technology (2010)
E.V. Doktorov, S.B. Leble, Dressing Method in Mathematical Physics (Springer, Dordrecht, 2007)
IYu. Popov, S.L. Popova, Eigenvalues and bands imbedded in the continuous spectrum for a system of resonators and a waveguide: solvable model. Phys. Lett. A 222, 286–290 (1996)
V. Fock, Fundamentals of Quantum Mechanics. (Mir publishers, Moscow, 1978, 1982)
R. Szmytkowski, Zero-range potentials for Dirac particles: scattering and related continuum problems. Phys. Rev. A 71(052708), 1–19 (2005)
S. Botman, S. Leble, Bloch wave scattering on pseudopotential impurity in 1D Dirac comb model. arXiv:1511.04758v1 [cond-mat.mes-hall]
M. Le Bellac, Quantum Physics (Cambridge University Press, Cambridge, 2006)
C.A. Coulson, B. O’Leary, R.B. Mallion, Hückel Theory for Organic Chemists (Academic Press, London, 1978)
M. Nakahara, C. Wakai, N. Matubayasi, Jump in the rotational mobility of Benzene induced by the Clathrate Hydrate formation. J. Phys. Chem. 99(5), 1377–1379 (1995)
B.S. Pavlov, A.V. Strepetov, Exactly solvable model of electron scattering by an inhomogeneity in a thin conductor. TMF 90(2), 226–232 (1992); Theoret. Math. Phys. 90 (2), 152–156 (1992)
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Leble, S. (2019). Nanowaveguides. Bloch Waves. In: Waveguide Propagation of Nonlinear Waves. Springer Series on Atomic, Optical, and Plasma Physics, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-030-22652-7_8
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