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
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TY - JOUR
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→∞
UR - http://www.scopus.com/inward/record.url?scp=54049088497&partnerID=8YFLogxK
U2 - 10.1007/s00208-008-0264-x
DO - 10.1007/s00208-008-0264-x
M3 - Journal article
AN - SCOPUS:54049088497
VL - 343
SP - 35
EP - 51
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 1
ER -
ID: 201991421