Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zeta-regularized determinant

JOURNAL OF GEOMETRY AND PHYSICS

卷:60

Article

Bauer, Wolfram¹  Furutani, Kenro² 

[1] Univ Gottingen, Math Inst, D-37073 Gottingen, Germany

[2] Tokyo Univ Sci, Fac Sci & Technol, Dept Math, Chiba 2788510, Japan

关键词: Sub-Laplacian;    Heat kernel;    Spectral zeta function;    Heisenberg manifold;    Kodaira-Thurston manifold;    Zeta-regularized determinant;   

DOI  :  10.1016/j.geomphys.2010.04.009

来源: Elsevier

【 摘 要 】

We analyze the spectral zeta function for sub-Laplace operators on product manifolds M x N. Starting from suitable conditions on the zeta functions on each factor, the existence of a meromorphic extension to the complex plane and real analyticity in a zero neighbourhood is proved. In the special case of N = S(1) and using the Poisson summation formula, we obtain expressions for the zeta-regularized determinant. Moreover, we can calculate limit cases of such determinants by inserting a parameter into our formulas. This is a generalization of results in Furutani and de Gosson (2003) [1] and in particular it applies to an intrinsic sub-Laplacian on U(2) congruent to S(3) x S(1) induced by a sum of squares of canonical vector fields on S(3); cf. Bauer and Furutani (2008) [2]. Finally, the spectral zeta function of a sub-Laplace operator on Heisenberg manifolds is calculated by using an explicit expression of the heat kernel for the corresponding sub-Laplace operator on the Heisenberg group; cf. Beals et al. (2000) [18] and Hulanicki (1976) [19]. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

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