RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Zametki, 2009, Volume 86, Issue 6, Pages 819–828 (Mi mz6575)  

This article is cited in 7 scientific papers (total in 7 papers)

Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions

V. V. Grushin

Moscow State Institute of Electronics and Mathematics

Abstract: In the present paper, a family of linear Fredholm operators depending on several parameters is considered. We implement a general approach, which allows us to reduce the problem of finding the set $\Lambda$ of parameters $t=(t_1,…,t_n)$ for which the equation $A(t)u=0$ has a nonzero solution to a finite-dimensional case. This allows us to obtain perturbation theory formulas for simple and conic points of the set $\Lambda$ by using the ordinary implicit function theorems. These formulas are applied to the existence problem for the conic points of the eigenvalue set $E(k)$ in the space of Bloch functions of the two-dimensional Schrödinger operator with a periodic potential with respect to a hexagonal lattice.

Keywords: multiparameter perturbation theory, Fredholm operator, hexagonal lattice, Bloch function, two-dimensional Schrödinger operator, Hilbert space, analytic function

DOI: https://doi.org/10.4213/mzm6575

Full text: PDF file (465 kB)
References: PDF file   HTML file

English version:
Mathematical Notes, 2009, 86:6, 767–774

Bibliographic databases:

UDC: 517.958
Received: 11.11.2008

Citation: V. V. Grushin, “Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions”, Mat. Zametki, 86:6 (2009), 819–828; Math. Notes, 86:6 (2009), 767–774

Citation in format AMSBIB
\Bibitem{Gru09}
\by V.~V.~Grushin
\paper Multiparameter Perturbation Theory of Fredholm Operators Applied to Bloch Functions
\jour Mat. Zametki
\yr 2009
\vol 86
\issue 6
\pages 819--828
\mathnet{http://mi.mathnet.ru/mz6575}
\crossref{https://doi.org/10.4213/mzm6575}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2643450}
\zmath{https://zbmath.org/?q=an:1197.47025}
\elib{http://elibrary.ru/item.asp?id=15296228}
\transl
\jour Math. Notes
\yr 2009
\vol 86
\issue 6
\pages 767--774
\crossref{https://doi.org/10.1134/S0001434609110194}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000273362000019}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-73949096433}


Linking options:
  • http://mi.mathnet.ru/eng/mz6575
  • https://doi.org/10.4213/mzm6575
  • http://mi.mathnet.ru/eng/mz/v86/i6/p819

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. J. Brüning, V. V. Grushin, S. Yu. Dobrokhotov, “Averaging of Linear Operators, Adiabatic Approximation, and Pseudodifferential Operators”, Math. Notes, 92:2 (2012), 151–165  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Fefferman Ch.L., Weinstein M.I., “Honeycomb lattice potentials and Dirac points”, J. Amer. Math. Soc., 25:4 (2012), 1169–1220  crossref  mathscinet  zmath  isi  elib  scopus
    3. Brüning J. Grushin V.V. Dobrokhotov S.Yu., “Approximate formulas for eigenvalues of the Laplace operator on a torus arising in linear problems with oscillating coefficients”, Russ. J. Math. Phys., 19:3 (2012), 261–272  crossref  mathscinet  zmath  isi  elib  scopus
    4. Kuchment P., “An overview of periodic elliptic operators”, Bull. Amer. Math. Soc., 53:3 (2016), 343–414  crossref  mathscinet  zmath  isi  elib  scopus
    5. Berkolaiko G., Comech A., “Symmetry and Dirac Points in Graphene Spectrum”, J. Spectr. Theory, 8:3 (2018), 1099–1147  crossref  mathscinet  zmath  isi  scopus
    6. Keller R.T., Marzuola J.L., Osting B., Weinstein I M., “Spectral Band Degeneracies of Pi/2-Rotationally Invariant Periodic Schrodinger Operators”, Multiscale Model. Simul., 16:4 (2018), 1684–1731  crossref  isi
    7. Lee-Thorp J.P., Weinstein M.I., Zhu Y., “Elliptic Operators With Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene”, Arch. Ration. Mech. Anal., 232:1 (2019), 1–63  crossref  mathscinet  zmath  isi  scopus
  • Математические заметки Mathematical Notes
    Number of views:
    This page:346
    Full text:80
    References:44
    First page:10

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019