Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture

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Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture. / Møller, Niels Martin.

I: Mathematische Annalen, Bind 343, Nr. 1, 01.01.2009, s. 35-51.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Møller, NM 2009, 'Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture', Mathematische Annalen, bind 343, nr. 1, s. 35-51. https://doi.org/10.1007/s00208-008-0264-x

APA

Møller, N. M. (2009). Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture. Mathematische Annalen, 343(1), 35-51. https://doi.org/10.1007/s00208-008-0264-x

Vancouver

Møller NM. Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture. Mathematische Annalen. 2009 jan. 1;343(1):35-51. https://doi.org/10.1007/s00208-008-0264-x

Author

Møller, Niels Martin. / Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture. I: Mathematische Annalen. 2009 ; Bind 343, Nr. 1. s. 35-51.

Bibtex

@article{5a2c3de6ead64e1980378b9b86ecf69b,
title = "Dimensional asymptotics of determinants on S n , and proof of B{\"a}r-Schopka's conjecture",
abstract = "We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians Δ + β R on round S n . For Laplacians the behavior depends on {"}the coupling strength{"} β, and one cannot in general expect a finite limit of ζ′(0), and for the ordinary Laplacian, β = 0, we prove it to be +∞, for odd dimensions. For the Dirac operator, B{\"a}r and Schopka conjectured a limit of unity for the determinant (B{\"a}r and Schopka, Geometric Analysis and Nonlinear PDEs, pp. 39-67, 2003), i.e. lim det (D, Scan n)=1. n→∞ We prove their conjecture rigorously, giving asymptotics, as well as a pattern of inequalities satisfied by the determinants. The limiting value of unity is a virtue of having {"}enough scalar curvature{"} and no kernel. Thus, for the important (conformally covariant) Yamabe operator, β = (n-2)/(4(n-1)), the determinant tends to unity. For the ordinary Laplacian it is natural to rescale spheres to unit volume, since lim det(Δ, Srescaled 2k+1)=1/2πe. k→∞",
author = "M{\o}ller, {Niels Martin}",
year = "2009",
month = jan,
day = "1",
doi = "10.1007/s00208-008-0264-x",
language = "English",
volume = "343",
pages = "35--51",
journal = "Mathematische Annalen",
issn = "0025-5831",
publisher = "Springer",
number = "1",

}

RIS

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T1 - Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture

AU - Møller, Niels Martin

PY - 2009/1/1

Y1 - 2009/1/1

N2 - We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians Δ + β R on round S n . For Laplacians the behavior depends on "the coupling strength" β, and one cannot in general expect a finite limit of ζ′(0), and for the ordinary Laplacian, β = 0, we prove it to be +∞, for odd dimensions. For the Dirac operator, Bär and Schopka conjectured a limit of unity for the determinant (Bär and Schopka, Geometric Analysis and Nonlinear PDEs, pp. 39-67, 2003), i.e. lim det (D, Scan n)=1. n→∞ We prove their conjecture rigorously, giving asymptotics, as well as a pattern of inequalities satisfied by the determinants. The limiting value of unity is a virtue of having "enough scalar curvature" and no kernel. Thus, for the important (conformally covariant) Yamabe operator, β = (n-2)/(4(n-1)), the determinant tends to unity. For the ordinary Laplacian it is natural to rescale spheres to unit volume, since lim det(Δ, Srescaled 2k+1)=1/2πe. k→∞

AB - We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians Δ + β R on round S n . For Laplacians the behavior depends on "the coupling strength" β, and one cannot in general expect a finite limit of ζ′(0), and for the ordinary Laplacian, β = 0, we prove it to be +∞, for odd dimensions. For the Dirac operator, Bär and Schopka conjectured a limit of unity for the determinant (Bär and Schopka, Geometric Analysis and Nonlinear PDEs, pp. 39-67, 2003), i.e. lim det (D, Scan n)=1. n→∞ We prove their conjecture rigorously, giving asymptotics, as well as a pattern of inequalities satisfied by the determinants. The limiting value of unity is a virtue of having "enough scalar curvature" and no kernel. Thus, for the important (conformally covariant) Yamabe operator, β = (n-2)/(4(n-1)), the determinant tends to unity. For the ordinary Laplacian it is natural to rescale spheres to unit volume, since lim det(Δ, Srescaled 2k+1)=1/2πe. k→∞

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U2 - 10.1007/s00208-008-0264-x

DO - 10.1007/s00208-008-0264-x

M3 - Journal article

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SP - 35

EP - 51

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

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