Geodesic
The Green's function in the problem of charge dynamics on a one-dimensional lattice with an impurity center
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 1, pp. 88-98 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the “tight-binding” approximation (the Hückel model), we consider the evolution of the charge wave function on a semi-infinite one-dimensional lattice with an additional energy $U$ at a single impurity site. In the case of the continuous spectrum (for $|U|<1)$ where there is no localized state, we construct the Green's function using the expansion in terms of eigenfunctions of the continuous spectrum and obtain an expression for the time Green's function in the form of a power series in $U$. It unexpectedly turns out that this series converges absolutely even in the case where the localized state is added to the continuous spectrum. We can therefore say that the Green's function constructed using the states of the continuous spectrum also contains an implicit contribution from the localized state.
Keywords: Green's function, quantum dynamics. 
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V. N. Likhachev; G. A. Vinogradov. The Green's function in the problem of charge dynamics on a one-dimensional lattice with an impurity center. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 1, pp. 88-98. http://geodesic.mathdoc.fr/item/TMF_2018_196_1_a7/

[1] G. Cuniberti, G. Fagas, K. Richter, “Fingerprints of mesoscopic leads in the conductance of a molecular wire”, Chem. Phys., 281:2–3 (2002), 465-476 | DOI

[2] T. N. Todorov, “Tight-binding simulation of current-carrying nanostructures”, J. Phys.: Condens. Matter, 14:11 (2002), 3049–3084 | DOI

[3] R. Movassagh, G. Strang, Y. Tsuji, R. Hoffman, “The Green's function for the Hückel (tight binding) model”, J. Math. Phys., 58:3 (2017), 033505, 20 pp. | DOI | MR

[4] K. D. Sattler (ed.), Fundamentals of Picoscience, CRC Press, Boca Raton, FL, 2014

[5] D. A. Ryndyk, R. Gutiérrez, B. Song, G. Cuniberti, “Green function techniques in the treatment of quantum transport at the molecular scale”, Energy Transfer Dynamics in Biomaterial Systems, Springer Series in Chemical Physics, 93, eds. I. Burghardt, V. May, D. A. Micha, E. R. Bittner, Springer, Berlin, Heidelberg, 2009, 213–335 | DOI

[6] Yu. A. Kruglyak, Nanoelektronika “snizu – vverkh”, Izd-vo Strelbitskogo, Kiev, 2016

[7] V. N. Likhachev, T. Yu. Astakhova, G. A. Vinogradov, “‘Elektronnyi ping-pong’ na odnomernoi reshetke. Dvizhenie volnovogo paketa do pervogo otrazheniya”, TMF, 175:2 (2013), 279–299 | DOI | DOI | MR | Zbl

[8] V. N. Likhachev, T. Yu. Astakhova, G. A. Vinogradov, “‘Elektronnyi ping-pong’ na odnomernoi reshetke. Mnogokratnye otrazheniya volnovogo paketa i zakhvat volnovoi funktsii aktseptorom”, TMF, 176:2 (2013), 306–321 | DOI | DOI | MR | Zbl

[9] V. N. Likhachev, O. I. Shevaleevskii, G. A. Vinogradov, “Quantum dynamics of charge transfer on the one-dimensional lattice: wave packet spreading and recurrence”, Chinese Phys. B, 25:1 (2016), 018708 | DOI

[10] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1962 | MR | Zbl

[11] N. J. Balmforth, P. J. Morrison, “Normal modes and continuous spectra”, Ann. N. Y. Acad. Sci., 773, no. 1, 1995, 80–94 | DOI

[12] R. O. Kuzmin, Besselevy funktsii, ONTI, M.–L., 1935