Russian Mathematical Surveys
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






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


Russian Mathematical Surveys, 2020, Volume 75, Issue 1, Pages 147–186
DOI: https://doi.org/10.1070/RM9917
(Mi rm9917)
 

This article is cited in 3 scientific papers (total in 3 papers)

Etudes of the resolvent

L. A. Takhtajanab

a The Euler International Mathematical Institute
b Stony Brook University, Stony Brook, New York
References:
Abstract: Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number theory from a unified point of view. For instance, for the Sturm–Liouville operator there is a derivation of the Gelfand–Levitan trace formula, and for the one-dimensional Schrödinger operator a derivation of Faddeev's formula for the characteristic determinant and the Zakharov–Faddeev trace identities. Recent results on the spectral theory of a certain functional-difference operator arising in conformal field theory are then presented. The last section of the survey is devoted to the Laplace operator on a fundamental domain of a Fuchsian group of the first kind on the Lobachevsky plane. An algebraic scheme is given for proving analytic continuation of the integral kernel of the resolvent of the Laplace operator and the Eisenstein–Maass series. In conclusion there is a discussion of the relationship between the values of the Eisenstein–Maass series at Heegner points and the Dedekind zeta-functions of imaginary quadratic fields, and it is explained why pseudo-cusp forms for the case of the modular group do not provide any information about the zeros of the Riemann zeta-function.
Bibliography: 50 titles.
Keywords: resolvent of an operator, characteristic determinant of an operator, Hilbert identities, Sturm–Liouville operator, Gelfand–Levitan trace formula, Schrödinger operator, functional-difference operator, Laplace operator on the Lobachevsky plane, eigenfunction expansions, Jost solutions, Zakharov–Faddeev trace identities, Eisenstein–Maass series, ,Dedekind zeta-functions of imaginary quadratic fields, Riemann zeta-function.
Received: 03.10.2019
Bibliographic databases:
Document Type: Article
UDC: 517.984+511.384.4
MSC: Primary 47A10, 47B25; Secondary 34B24, 35P10, 47E05
Language: English
Original paper language: Russian
Citation: L. A. Takhtajan, “Etudes of the resolvent”, Russian Math. Surveys, 75:1 (2020), 147–186
Citation in format AMSBIB
\Bibitem{Tak20}
\by L.~A.~Takhtajan
\paper Etudes of the resolvent
\jour Russian Math. Surveys
\yr 2020
\vol 75
\issue 1
\pages 147--186
\mathnet{http://mi.mathnet.ru//eng/rm9917}
\crossref{https://doi.org/10.1070/RM9917}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4070021}
\zmath{https://zbmath.org/?q=an:1440.47004}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2020RuMaS..75..147T}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000613181400001}
\elib{https://elibrary.ru/item.asp?id=43714800}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85085031540}
Linking options:
  • https://www.mathnet.ru/eng/rm9917
  • https://doi.org/10.1070/RM9917
  • https://www.mathnet.ru/eng/rm/v75/i1/p155
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
    Statistics & downloads:
    Abstract page:879
    Russian version PDF:208
    English version PDF:44
    References:75
    First page:73
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024