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



Izv. RAN. Ser. Mat.:
Year:
Volume:
Issue:
Page:
Find






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


Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 5, Pages 139–176 (Mi izv4258)  

On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains

D. A. Popov

A. N. Belozersky Institute of Physico-Chemical Biology, M. V. Lomonosov Moscow State University

Abstract: We construct Liouville metrics on the two-dimensional torus for which the asymptotic behaviour of the second term in the Weyl formula is evaluated explicitly. We prove the instability of the second term in this formula with respect to small deformations (in the $C^1$ metric) of a Liouville metric, and establish the absence of power reduction in the Hörmander estimate on the class of closed manifolds with smooth metric in the case of integrable geodesic flow and the zero measure of the set of closed geodesics in the subspace of unit spheres of the cotangent bundle.

Keywords: Laplace operator, spectrum, Weyl formula, integer points, geodesic flow.

DOI: https://doi.org/10.4213/im4258

Full text: PDF file (726 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2011, 75:5, 1007–1045

Bibliographic databases:

UDC: 517.984.5+511.338
MSC: Primary 11P21, 35P20; Secondary 35J05, 47F05, 58G50
Received: 18.11.2009

Citation: D. A. Popov, “On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains”, Izv. RAN. Ser. Mat., 75:5 (2011), 139–176; Izv. Math., 75:5 (2011), 1007–1045

Citation in format AMSBIB
\Bibitem{Pop11}
\by D.~A.~Popov
\paper On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains
\jour Izv. RAN. Ser. Mat.
\yr 2011
\vol 75
\issue 5
\pages 139--176
\mathnet{http://mi.mathnet.ru/izv4258}
\crossref{https://doi.org/10.4213/im4258}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2884666}
\zmath{https://zbmath.org/?q=an:05988246}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011IzMat..75.1007P}
\elib{http://elibrary.ru/item.asp?id=20358813}
\transl
\jour Izv. Math.
\yr 2011
\vol 75
\issue 5
\pages 1007--1045
\crossref{https://doi.org/10.1070/IM2011v075n05ABEH002562}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296665700007}
\elib{http://elibrary.ru/item.asp?id=18012407}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80555131757}


Linking options:
  • http://mi.mathnet.ru/eng/izv4258
  • https://doi.org/10.4213/im4258
  • http://mi.mathnet.ru/eng/izv/v75/i5/p139

    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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
    Number of views:
    This page:443
    Full text:92
    References:55
    First page:15

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