TY - JOUR

T1 - Spectral zeta function of a sub-Laplacian on product sub-Riemannian manifolds and zeta-regularized determinant

AU - Bauer, Wolfram

AU - Furutani, Kenro

N1 - Funding Information: The first author has been supported by an “Emmy-Noether scholarship” of DFG ( Deutsche Forschungsgemeinschaft ). The second author has been partially supported by the Grant-in-aid for Scientific Research (C) No. 20540218 , Japan Society for the Promotion of Science . Copyright: Copyright 2010 Elsevier B.V., All rights reserved.

PY - 2010/4/27

Y1 - 2010/4/27

N2 - We analyze the spectral zeta function for sub-Laplace operators on product manifolds M×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=S1 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)≅S3×S1 induced by a sum of squares of canonical vector fields on S3; 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].

AB - We analyze the spectral zeta function for sub-Laplace operators on product manifolds M×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=S1 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)≅S3×S1 induced by a sum of squares of canonical vector fields on S3; 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].

KW - Heat kernel

KW - Heisenberg manifold

KW - Kodaira-Thurston manifold

KW - Spectral zeta function

KW - Sub-Laplacian

KW - Zeta-regularized determinant

UR - http://www.scopus.com/inward/record.url?scp=77953137014&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2010.04.009

DO - 10.1016/j.geomphys.2010.04.009

M3 - Article

AN - SCOPUS:77953137014

VL - 60

SP - 1209

EP - 1234

JO - Journal of geometry and physics

JF - Journal of geometry and physics

SN - 0393-0440

IS - 9

ER -