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Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 7
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Articles

WKB asymptotics of meromorphic solutions to difference equations

Alexander Fedotova St. Petersburg State University, St.Petersburg, RussiaCorrespondence[email protected]
ORCID Iconhttps://orcid.org/0000-0002-8010-3004View further author information
ORCID Icon &
Frédéric Kloppb Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu – Paris Rive Gauche, Paris, FranceView further author information
Pages 1557-1573 | Received 19 Apr 2019, Accepted 26 Jul 2019, Published online: 16 Aug 2019

References

  • Fedoryuk A. Asymptotic analysis. Linear ordinary differential equations. Berlin, Heidelberg: Springer-Verlag; 2009.
  • Wasow W. Asymptotic expansions for ordinary differential equations. New York: Dover Publications; 1987.
  • Buslaev V, Fedotov A. Complex WKB method for Harper equation. St Petersburg Math J. 1995;6(3):495–517.
  • Fedotov A, Shchetka E. Complex WKB method for the difference Schrödinger equation with the potential being a trigonometric polynomial. St Petersburg Math J. 2018;29:363–381. doi: 10.1090/spmj/1497
  • Fedotov A, Klopp F. The complex WKB method for difference equations and Airy functions. arXiv:1810.04918.
  • Wilkinson M. An exact renormalisation group for Bloch electrons in a magnetic field. J Phys A: Math Gen. 1987;20:4337–4354. doi: 10.1088/0305-4470/20/13/035
  • Helffer B, Sjöstrand J. Analyse semi-classique pour l'équation de Harper (avec application à l'équationde Schrödinger avec champ magnétique). Mémoires de la SMF (nouv. sér.).. 1988;34:1–113.
  • Guillement JP, Helffer B, Treton P. Walk inside Hofstadter's butterfly. J Phys France. 1989;50:2019–2058. doi: 10.1051/jphys:0198900500150201900
  • Avila A, Jitomirskaya S. The Ten Martini Problem. Ann Math. 2009;170:303–342. doi: 10.4007/annals.2009.170.303
  • Grempel D, Fishman S, Prange R. Localization in an incommensurate potential: An exactly solvable model. Phys Rev Lett. 1982;49:833–836. doi: 10.1103/PhysRevLett.49.833
  • Babich V, Lyalinov M, Grikurov V. Diffraction theory: the Sommerfeld-Malyuzhinets technique. Oxford: Alpha Science; 2008.
  • Fedotov A, Klopp F. Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case. Commun Math Phys. 2002;227:1–92. doi: 10.1007/s002200200612
  • Geronimo JS, Bruno O, Van AsscheW. WKB and turning point theory for second-order difference equations. Oper Theory Adv App. 1992;69:269–301. doi: 10.1016/0021-9045(92)90003-7
  • Dobrokhotov SYu, Tsvetkova AV. On lagrangian manifolds related to asymptotics of Hermite polynomials. Math Notes. 2018;104(6):810–822. doi: 10.1134/S0001434618110263
  • Fedotov A. Monodromization method in the theory of almost-periodic equations. St Petersburg Math J. 2014;25:303–325. doi: 10.1090/S1061-0022-2014-01292-7
  • Fedotov A, Sandomirskiy F. An exact renormalization formula for the Maryland model. Commun Math Phys. 2015;334(2):1083–1099. doi: 10.1007/s00220-014-2126-6
  • Jitomirskaya S, Liu W. Universal hierarchical structure of quasiperiodic eigenfunctions. Ann Math. 2018;187(3):721–776. doi: 10.4007/annals.2018.187.3.3
  • Fedotov A, Klopp F. Quasiclassical asymptotics of solutions to difference equations with meromorphic coefficients. Proceedings of the conference Days on Diffraction 2017. St. Petersburg: IEEE; 2017. pp. 110–111.
  • Fedotov A, Shchetka E. A complex WKB method for difference equations in bounded domains. J Math Sci. 2017;224:157–169. doi: 10.1007/s10958-017-3402-8
  • Fedotov A, Klopp F. On the absolutely continuous spectrum of an one-dimensional quasi-periodic Schrödinger operator in adiabatic limit. Trans AMS. 2005;357:4481–4516. doi: 10.1090/S0002-9947-05-03961-9

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