Geodesic
Semiclassical asymptotic behavior of the lower spectral bands of the Schrödinger operator with a trigonal-symmetric periodic potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 2, pp. 264-277 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the semiclassical approximation of the lower bands of the Schrödinger operator with a periodic two-dimensional potential with a trigonal symmetry and consider the cases where the potential has one or two wells in the elementary cell. We obtain the exponentially small asymptotic behavior of the band width and find the dispersion relations. We investigate the form of the Bloch functions. Solving this problem is the first step in studying the more complicated (and more physically interesting) problem of tunnel effects in rotating dimers.
Keywords: periodic Schrödinger operator, semiclassical asymptotic behavior, spectral band
Mots-clés : tunnel effect. 
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A. Yu. Anikin; M. A. Vavilova. Semiclassical asymptotic behavior of the lower spectral bands of the Schrödinger operator with a trigonal-symmetric periodic potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 2, pp. 264-277. http://geodesic.mathdoc.fr/item/TMF_2020_202_2_a5/

[1] M. I. Katsnelson, Graphene. Carbon in Two Dimensions, Cambridge Univ. Press, Cambridge, 2012

[2] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982 | MR | MR | Zbl

[3] B. Simon, “Semiclassical analysis of low lying eigenvalues. III. Width of the ground state band in strongly coupled solids”, Ann. Phys., 158:2 (1984), 415–420 | DOI | MR

[4] A. Outassourt, “Comportement semi-classique pour l'opérateur de Schrödinger à potentiel périodique”, J. Func. Anal., 72:1 (1987), 65–93 | DOI | MR

[5] B. Helffer, J. Sjöstrand, “Puits multiples en limite semi-classique. II. Interaction moléculaire. Symétries. Perturbation”, Ann. Inst. H. Poincaré. Phys. Théor., 42:2 (1985), 127–212 | MR | Zbl

[6] B. Helffer, J. Sjöstrand, “Multiple wells in the semi-classical limit I”, Comm. Partial Differ. Equ., 9:4 (1984), 337–408 | DOI | MR

[7] V. P. Maslov, “Globalnaya eksponentsialnaya asimptotika reshenii tunnelnykh uravnenii i zadachi o bolshikh ukloneniyakh”, Tr. MIAN SSSR, 163 (1984), 150–180 ; С. Ю. Доброхотов, В. Н. Колокольцов, В. П. Маслов, “Расщепление нижних энергетических уровней уравнения Шредингера и асимптотика фундаментального решения уравнения $hu_t=h^2\Delta u/2-V(x)u$”, ТМФ, 87:3 (1991), 323–375 | MR | Zbl | DOI | MR | Zbl

[8] S. Yu. Dobrokhotov, V. N. Kolokoltsov, “Ob amplitude rasschepleniya nizhnikh energeticheskikh urovnei operatora Shredingera s dvumya simmetrichnymi yamami”, TMF, 94:3 (1993), 426–434 | DOI | MR | Zbl

[9] B. Simon, “Semiclassical analysis of low lying eigenvalues. II. Tunneling”, Ann. Math., 120:1 (1984), 89–118 | DOI | MR

[10] J. Brüning, S. Yu. Dobrokhotov, E. S. Semenov, “Unstable closed trajectories, librations and splitting of the lowest eigenvalues in quantum double well problem”, Regul. Chaotic Dyn., 11:2 (2006), 167–180 | DOI | MR | Zbl

[11] A. Yu. Anikin, “Asymptotic behaviour of the Maupertuis action on a libration and a tunneling in a double well”, Rus. J. Math. Phys., 20:1 (2013), 1–10 | MR

[12] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. 3, Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1974 ; Е. М. Лифшиц, Л. П. Питаевский, Теоретическая физика, т. 9, Статистическая физика. Часть 2. Теория конденсированного состояния, Физматлит, М., 2004 | MR | MR

[13] E. M. Harrell, “On the rate of asymptotic eigenvalue degeneracy”, Commun. Math. Phys., 60:1 (1978), 73–95 ; “Double wells”, Commun. Math. Phys., 75:3 (1980), 239–261 | DOI | MR | DOI | MR

[14] E. M. Harrell, “The band-structure of a one-dimensional periodic system in a scaling limit”, Ann. Phys., 119:2 (1979), 351–369 | DOI | MR

[15] A. Yu. Anikin, S. Yu. Dobrokhotov, M. I. Katsnelson, “Nizhnyaya chast spektra dvumernogo operatora Shredingera s periodicheskim po odnoi peremennoi potentsialom i prilozheniya k kvantovym dimeram”, TMF, 188:2 (2016), 288–317 | DOI | DOI | MR

[16] C. Fusco, A. Fasolino, T. Janssen, “Nonlinear dynamics of dimers on periodic substrates”, Eur. Phys. J. B, 31:1 (2003), 95–102, arXiv: ; E. Pijper, A. Fasolino, “Mechanisms for correlated surface diffusion of weakly bonded dimers”, Phys. Rev. B, 72:16 (2005), 165328, 5 pp. cond-mat/0301357 | DOI | DOI

[17] V. V. Kozlov, S. V. Bolotin, “Libratsiya v sistemakh so mnogimi stepenyami svobody”, PMM, 42:2 (1978), 245–250 ; “Об асимптотических решениях уравнений динамики”, Вестн. Моск. ун-та. Сер. 1. Матем., мех., 1980, No 4, 84–89 | DOI | MR | Zbl | MR | Zbl