A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds
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Author(s)
Fan, Colin
Kim, Elena
Zeytuncu, Yunus E.
Date Issued
February 14, 2022
Publisher
Springer International Publishing
Citation
Complex Analysis and its Synergies. 2022 Feb 14;8(1):4
Version
Author's final manuscript
Abstract
Abstract
Let
$$M= \Gamma \setminus \mathbb {H}_d$$
M
=
Γ
\
H
d
be a compact quotient of the d-dimensional Heisenberg group
$$\mathbb {H}_d$$
H
d
by a lattice subgroup
$$\Gamma $$
Γ
. We show that the eigenvalue counting function
$$N^\alpha \left( \lambda \right) $$
N
α
λ
for any fixed element of a family of second order differential operators
$$\left\{ \mathcal {L}_\alpha \right\} $$
L
α
on M has asymptotic behavior
$$N^\alpha \left( \lambda \right) \sim C_{d,\alpha } {\text {vol}}\left( M\right) \lambda ^{d + 1}$$
N
α
λ
∼
C
d
,
α
vol
M
λ
d
+
1
, where
$$C_{d,\alpha }$$
C
d
,
α
is a constant that only depends on the dimension d and the parameter
$$\alpha $$
α
. As a consequence, we obtain an analog of Weyl’s law (both on functions and forms) for the Kohn Laplacian on M. Our main tools are Folland’s description of the spectrum of
$${\mathcal {L}}_{\alpha }$$
L
α
and Karamata’s Tauberian theorem.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s40627-022-00094-3
