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St. Petersburg Mathematical Journal

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Complex WKB method for a difference Schrödinger equation with the potential being a trigonometric polynomial


Authors: A. Fedotov and E. Shchetka
Translated by: A. Fedotov
Original publication: Algebra i Analiz, tom 29 (2017), nomer 2.
Journal: St. Petersburg Math. J. 29 (2018), 363-381
MSC (2010): Primary 34E20
DOI: https://doi.org/10.1090/spmj/1497
Published electronically: March 12, 2018
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Abstract | References | Similar Articles | Additional Information

Abstract: The difference Schrödinger equation $ \psi (z+h)+\psi (z-h)+ v(z)\psi (z)=E\psi (z)$, $ z\in \mathbb{C}$, is considered; here $ h$ is a positive number, $ E$ is the spectral parameter, and $ v$ is a trigonometric polynomial. The asymptotics of its entire solutions is studied as $ h\to 0$.


References [Enhancements On Off] (What's this?)

  • 1. V. M. Babich, M. A. Lyalinov, and V. E. Grikurov, Diffraction theory. The Sommerfeld-Malyuzhinets technique, Alpha Sci. Ser. Wave Phenom., Alpha Sci. Ltd, Oxford, 2008.
  • 2. V. S. Buslaev and A. A. Fedotov, The complex WKB method for the Harper equation, Algebra i Analiz 6 (1994), no. 3, 59–83 (Russian); English transl., St. Petersburg Math. J. 6 (1995), no. 3, 495–517. MR 1301830
  • 3. J. P. Guillement, B. Helffer, and P. Treton, Walk inside Hofstadter's butterfly, J. Phys. France 50 (1989), 2019-2058.
  • 4. Alexander Fedotov and Frédéric Klopp, A complex WKB method for adiabatic problems, Asymptot. Anal. 27 (2001), no. 3-4, 219–264 (English, with English and French summaries). MR 1858917
  • 5. M. V. Fedoryuk, \cyr Asimptoticheskie metody dlya lineĭnykh obyknovennykh differentsial′nykh uravneniĭ, \cyr Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1983 (Russian). MR 732787
    Mikhail V. Fedoryuk, Asymptotic analysis, Springer-Verlag, Berlin, 1993. Linear ordinary differential equations; Translated from the Russian by Andrew Rodick. MR 1295032
  • 6. A. A. Fedotov and E. V. Schetka, A complex WKB method for difference equations in bounded domains, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 438 (2015), no. Matematicheskie Voprosy Teorii Rasprostraneniya Voln. 45, 236–254 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 224 (2017), no. 1, 157–169. MR 3501077

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Additional Information

A. Fedotov
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, Russia
Email: a.fedotov@spbu.ru

E. Shchetka
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, Russia
Email: shchetka.ekaterina@mail.ru

DOI: https://doi.org/10.1090/spmj/1497
Keywords: Difference Schr\"odinger equation, complex WKB method
Received by editor(s): October 10, 2016
Published electronically: March 12, 2018
Additional Notes: Supported by RFBR (grant no. 17-01-00668-a) and by the “Möbius Contest”, a foundation for support of young scientists.
Dedicated: Dedicated to the memory of Vladimir Savel’evich Buslaev
Article copyright: © Copyright 2018 American Mathematical Society

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