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Zeta regularized products and functional determinants on spheres. (English) Zbl 0864.47024
The authors use a factorization theorem for zeta regularized products to compute the functional determinant of the Laplacian on the sphere $S^n$ with the standard metric. They also determine the functional determinant of the conformal Laplacian on an even-dimensional sphere. The computations in this paper agree with those of {\it T. P. Branson} and {\it B. Ørsted} [Proc. Am. Math. Soc. 113, No. 3, 669-682 (1991; Zbl 0762.47019)]. The authors list the values of the functional determinant for the ordinary Laplacian in dimensions $n=2,3,4,5,6$ and for the conformal Laplacian in dimensions $4,6,8$.

47F05Partial differential operators
58J50Spectral problems; spectral geometry; scattering theory
Full Text: DOI Link
[1] E.W. Barnes, The theory of the multiple gamma function , Trans. Cambridge Philos. Soc. 19 (1904), 374-425. · Zbl 35.0462.01
[2] T.P. Branson, S-Y.A. Chang and P. Yang, Estimates and extremal problems for zeta function determinants on four-manifolds , · Zbl 0761.58053 · doi:10.1007/BF02097624
[3] T.P. Branson and B. Orsted, Explicit functional determinants in four dimensions , Proc. Amer. Math. Soc. 113 (1991), 669-682. JSTOR: · Zbl 0762.47019 · doi:10.2307/2048601 · http://links.jstor.org/sici?sici=0002-9939%28199111%29113%3A3%3C669%3AEFDIFD%3E2.0.CO%3B2-T&origin=euclid
[4] W.F. Donoghue, Jr., Distributions and Fourier transforms , Academic Press, New York, 1969. · Zbl 0188.18102
[5] D.B. Ray and I.M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds , Adv. Math. 7 (1974), 145-210. · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4
[6] B. Osgood, R. Philips and P. Sarnak, Extremals of determinants of Laplacians , J. Funct. Anal. 80 (1988), 148-211. · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5
[7] J.R. Quine, S.H. Heydari and R.Y. Song, Zeta regularized products , Trans. Amer. Math. Soc. 338 (1993), 213-231. · Zbl 0774.30030 · doi:10.2307/2154453
[8] I. Vardi, Determinants of Laplacians and multiple gamma functions , SIAM J. Math. Anal. 19 (1988), 493-507. · Zbl 0641.33003 · doi:10.1137/0519035
[9] --------, Computational recreations in Mathematica , Addison-Wesley, Redwood City, 1991. · Zbl 0786.11002
[10] M.F. Vignéras, L’Équation Fonctionelle de la Fonction Zeta de Selberg du Groupe Modulaire $PSL(2,Z)$, Astérisque 61 (1979), 235-249. · Zbl 0401.10036
[11] A. Voros, Spectral functions, special functions and the Selberg zeta function , Commun. Math. Phys. 110 (1987), 439-465. · Zbl 0631.10025 · doi:10.1007/BF01212422
[12] W.I. Weisberger, Normalization of the path integral measure and the coupling constants for bosonic strings , Nuclear Phys. B 284 (1987), 171-200. · doi:10.1016/0550-3213(87)90032-0