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

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

SN - 0025-5831

VL - 343

SP - 35

EP - 51

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1

ER -