Details
| Original language | English |
|---|---|
| Pages (from-to) | 242-261 |
| Number of pages | 20 |
| Journal | Journal des Mathematiques Pures et Appliquees |
| Volume | 97 |
| Issue number | 3 |
| Publication status | Published - 13 Jun 2011 |
| Externally published | Yes |
Abstract
We study the heat kernel trace and the spectral zeta function of an intrinsic sub-Laplace operator δ L\G sub on a two step compact nilmanifold L\G. Here G is an arbitrary nilpotent Lie group of step 2 and we assume the existence of a lattice L⊂ G. We essentially use the well-known heat kernel expressions of the sub-Laplacian on G due to Beals, Gaveau and Greiner. In contrast to the spectral zeta function of the Laplacian on L\. G which can have infinitely many simple poles it turns out that in case of the sub-Laplacian only one simple pole occurs. Its residue divided by the volume of L\. G is independent of L and can be expressed by the Lie group structure of G. By standard arguments this result is equivalent to a specific asymptotic behaviour of the heat kernel trace of δ L\G sub as time tends to zero. As an example we explicitly calculate the spectrum of the sub-Laplacian δ L\G sub in case of the six-dimensional free nilpotent Lie group G and a standard lattice L⊂ G by using a decomposition of δ L\G sub into a family of elliptic operators.
Keywords
- Hypoelliptic operator, Left-invariant Laplacian, Sub-elliptic heat kernel, Sub-Laplacian
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Journal des Mathematiques Pures et Appliquees, Vol. 97, No. 3, 13.06.2011, p. 242-261.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Spectral zeta function of the sub-Laplacian on two step nilmanifolds
AU - Bauer, W.
AU - Furutani, K.
AU - Iwasaki, C.
N1 - Funding Information: * Corresponding author. E-mail addresses: wbauer@uni-math.gwdg.de (W. Bauer), furutani_kenro@ma.noda.tus.ac.jp (K. Furutani), iwasaki@sci.u-hyogo.ac.jp (C. Iwasaki). 1 Supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungsgemeinschaft). 2 Partially supported by the Grant-in-Aid for Scientific Research (C) No. 20540218, Japan Society for the Promotion of Science. 3 Partially supported by the Grant-in-Aid for Scientific Research (C) No. 21540194, Japan Society for the Promotion of Science. Copyright: Copyright 2012 Elsevier B.V., All rights reserved.
PY - 2011/6/13
Y1 - 2011/6/13
N2 - We study the heat kernel trace and the spectral zeta function of an intrinsic sub-Laplace operator δ L\G sub on a two step compact nilmanifold L\G. Here G is an arbitrary nilpotent Lie group of step 2 and we assume the existence of a lattice L⊂ G. We essentially use the well-known heat kernel expressions of the sub-Laplacian on G due to Beals, Gaveau and Greiner. In contrast to the spectral zeta function of the Laplacian on L\. G which can have infinitely many simple poles it turns out that in case of the sub-Laplacian only one simple pole occurs. Its residue divided by the volume of L\. G is independent of L and can be expressed by the Lie group structure of G. By standard arguments this result is equivalent to a specific asymptotic behaviour of the heat kernel trace of δ L\G sub as time tends to zero. As an example we explicitly calculate the spectrum of the sub-Laplacian δ L\G sub in case of the six-dimensional free nilpotent Lie group G and a standard lattice L⊂ G by using a decomposition of δ L\G sub into a family of elliptic operators.
AB - We study the heat kernel trace and the spectral zeta function of an intrinsic sub-Laplace operator δ L\G sub on a two step compact nilmanifold L\G. Here G is an arbitrary nilpotent Lie group of step 2 and we assume the existence of a lattice L⊂ G. We essentially use the well-known heat kernel expressions of the sub-Laplacian on G due to Beals, Gaveau and Greiner. In contrast to the spectral zeta function of the Laplacian on L\. G which can have infinitely many simple poles it turns out that in case of the sub-Laplacian only one simple pole occurs. Its residue divided by the volume of L\. G is independent of L and can be expressed by the Lie group structure of G. By standard arguments this result is equivalent to a specific asymptotic behaviour of the heat kernel trace of δ L\G sub as time tends to zero. As an example we explicitly calculate the spectrum of the sub-Laplacian δ L\G sub in case of the six-dimensional free nilpotent Lie group G and a standard lattice L⊂ G by using a decomposition of δ L\G sub into a family of elliptic operators.
KW - Hypoelliptic operator
KW - Left-invariant Laplacian
KW - Sub-elliptic heat kernel
KW - Sub-Laplacian
UR - http://www.scopus.com/inward/record.url?scp=84856774992&partnerID=8YFLogxK
U2 - 10.1016/j.matpur.2011.06.003
DO - 10.1016/j.matpur.2011.06.003
M3 - Article
AN - SCOPUS:84856774992
VL - 97
SP - 242
EP - 261
JO - Journal des Mathematiques Pures et Appliquees
JF - Journal des Mathematiques Pures et Appliquees
SN - 0021-7824
IS - 3
ER -