Details
| Original language | English |
|---|---|
| Pages (from-to) | 1209-1234 |
| Number of pages | 26 |
| Journal | Journal of geometry and physics |
| Volume | 60 |
| Issue number | 9 |
| Publication status | Published - 27 Apr 2010 |
| Externally published | Yes |
Abstract
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].
Keywords
- Heat kernel, Heisenberg manifold, Kodaira-Thurston manifold, Spectral zeta function, Sub-Laplacian, Zeta-regularized determinant
ASJC Scopus subject areas
- Mathematics(all)
- Mathematical Physics
- Physics and Astronomy(all)
- Mathematics(all)
- Geometry and Topology
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In: Journal of geometry and physics, Vol. 60, No. 9, 27.04.2010, p. 1209-1234.
Research output: Contribution to journal › Article › Research › peer review
}
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 -