Geodesic
On monodromy matrices for a difference Schrödinger equation on the real line with a small periodic potential
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 223-244 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper one considers a one-dimensional difference Schrödinger equation $\psi(z+h) + \psi(z-h) + \lambda v(z) \psi(z) = E \psi(z) $ with a periodic potential $v$. In the case when the potential is real analytic, as well as in the case when, in a neighborhood of $\mathbb{R}$, the potential has a finite number of simple poles per period, for small values of the coupling constant $\lambda$, we describe the asymptotics of a monodromy matrix. 
@article{ZNSL_2021_506_a14,
     author = {K. S. Sedov and A. A. Fedotov},
     title = {On monodromy matrices for a difference {Schr\"odinger} equation on the real line with a small periodic potential},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {223--244},
     year = {2021},
     volume = {506},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a14/}
}
TY  - JOUR
AU  - K. S. Sedov
AU  - A. A. Fedotov
TI  - On monodromy matrices for a difference Schrödinger equation on the real line with a small periodic potential
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2021
SP  - 223
EP  - 244
VL  - 506
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a14/
LA  - ru
ID  - ZNSL_2021_506_a14
ER  - 
%0 Journal Article
%A K. S. Sedov
%A A. A. Fedotov
%T On monodromy matrices for a difference Schrödinger equation on the real line with a small periodic potential
%J Zapiski Nauchnykh Seminarov POMI
%D 2021
%P 223-244
%V 506
%U http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a14/
%G ru
%F ZNSL_2021_506_a14
K. S. Sedov; A. A. Fedotov. On monodromy matrices for a difference Schrödinger equation on the real line with a small periodic potential. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 223-244. http://geodesic.mathdoc.fr/item/ZNSL_2021_506_a14/

[1] M. Wilkinson, “Critical properties of electron eigenstates in incommensurate systems”, Proc. Roy. Soc. London Ser. A, 391 (1984), 305–350 | MR

[2] J. P. Guillement, B. Helffer, P. Treton, “Walk inside Hofstadter's butterfly”, J. Phys. France, 50 (1989), 2019–2058 | DOI

[3] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer-Verlag, Berlin, 1992 | Zbl

[4] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger Operators: with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, Berlin, 1987

[5] E. I. Dinaburg, Ya. G. Sinai, “Ob odnomernom uravnenii Shredingera s kvaziperiodicheskim potentsialom”, Funkts. analiz i ego prilozheniya, 9 (1975), 8–21

[6] J. Bellissard, R. Lima, D. Testard, “A metal-insulator transition for the almost Mathieu model”, Comm. Math. Phys., 88 (1983), 207–234 | DOI | MR | Zbl

[7] L. H. Eliasson, “Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation”, Comm. Math. Phys., 146 (1992), 447–482 | DOI | MR | Zbl

[8] A. Avila, S. Jitomirskaya, “Almost localization and almost reducibility”, J. Eur. Math. Soc., 12 (2010), 93–131 | DOI | Zbl

[9] D. Damanik, M. Goldstein, “On the inverse spectral problem for the quasi-periodic Schrödinger equation”, Publications Mathématiques de l'IHÉS, 119, 2014, 217–401 | DOI | Zbl

[10] B. Helffer, J. Sjostrand, “Analyse semi-classique pour l'équation de Harper”, Mém. Soc. Math. France (nouv. série), 34, 1988, 1–113

[11] A. A. Fedotov, “Metod monodromizatsii v teorii pochti-periodicheskikh uravnenii”, Algebra i analiz, 25 (2013), 203–235

[12] A. A. Fedotov, “Matritsa monodromii dlya uravneniya pochti-Mate s maloi konstantoi svyazi”, Funkts. analiz i ego prilozh., 52 (2018), 89–93 | Zbl

[13] A. Fedotov, A series of spectral gaps for the almost Mathieu operator with a small coupling constant, arXiv: 2012.03356 [math.SP]

[14] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, Mir, M., 1982

[15] A. A. Fedotov, “Monodromizatsiya i raznostnye uravneniya s meromorfnymi periodicheskimi koeffitsientami”, Funkts. analiz i ego prilozheniya, 52 (2018), 92–97 | Zbl

[16] Sriram Ganeshan, J. H. Pixley, S. Das Sarma, “Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension”, Phys. Rev. Lett., 114 (2015), 146601 | DOI

[17] V. S. Buslaev, A. A. Fedotov, “Uravnenie Kharpera: monodromizatsiya bez kvaziklassiki”, Algebra i analiz, 8 (1996), 65–97