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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 2(446), Pages 149–186 (Mi umn9870)  
This article is cited in 2 scientific papers (total in 2 papers)

Trace formula for the magnetic Laplacian

Yu. A. Kordyukovab, I. A. Taimanovcb

a Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa
b Novosibirsk State University
c Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: The Guillemin–Uribe trace formula is a semiclassical version of the Selberg trace formula and the more general Duistermaat–Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. This paper gives a survey of basic notions and results related to the Guillemin–Uribe trace formula and provides concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example.
Bibliography: 53 titles.

Keywords: trace formula, magnetic Laplacian, magnetic geodesics.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 14.Y26.31.0025
This work was supported by the Laboratory of Topology and Dynamics at Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).


DOI: https://doi.org/10.4213/rm9870

Full text: PDF file (756 kB)
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English version:
Russian Mathematical Surveys, 2019, 74:2, 325–361

Bibliographic databases:

UDC: 517.984+514.774.8
MSC: Primary 58J50; Secondary 37J35, 58J37, 81Q20
Received: 28.12.2018

Citation: Yu. A. Kordyukov, I. A. Taimanov, “Trace formula for the magnetic Laplacian”, Uspekhi Mat. Nauk, 74:2(446) (2019), 149–186; Russian Math. Surveys, 74:2 (2019), 325–361

Citation in format AMSBIB
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\pages 149--186
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    This publication is cited in the following articles:
    1. A. A. Ilyin, A. A. Laptev, “Magnetic Lieb–Thirring inequality for periodic functions”, Russian Math. Surveys, 75:4 (2020), 779–781  mathnet  crossref  crossref  mathscinet  isi  elib
    2. Yu. A. Kordyukov, I. A. Taimanov, “Quasi-classical approximation for magnetic monopoles”, Russian Math. Surveys, 75:6 (2020), 1067–1088  mathnet  crossref  crossref  mathscinet  isi  elib
    		• Успехи математических наук Russian Mathematical Surveys
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