A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds
Abstract Let $$M= \Gamma \setminus \mathbb {H}_d$$ M = Γ \...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2022
|
| Online Access: | https://hdl.handle.net/1721.1/140343 |
| _version_ | 1811071716063969280 |
|---|---|
| author | Fan, Colin Kim, Elena Zeytuncu, Yunus E. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Fan, Colin Kim, Elena Zeytuncu, Yunus E. |
| author_sort | Fan, Colin |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T08:55:20Z |
| format | Article |
| id | mit-1721.1/140343 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:55:20Z |
| publishDate | 2022 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1403432023-03-09T05:13:27Z A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds Fan, Colin Kim, Elena Zeytuncu, Yunus E. Massachusetts Institute of Technology. Department of Mathematics 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. 2022-02-15T12:58:37Z 2022-02-15T12:58:37Z 2022-02-14 2022-02-15T04:28:08Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/140343 Complex Analysis and its Synergies. 2022 Feb 14;8(1):4 en https://doi.org/10.1007/s40627-022-00094-3 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The Author(s), under exclusive licence to Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Fan, Colin Kim, Elena Zeytuncu, Yunus E. A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds |
| title | A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds |
| title_full | A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds |
| title_fullStr | A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds |
| title_full_unstemmed | A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds |
| title_short | A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds |
| title_sort | tauberian approach to an analog of weyl s law for the kohn laplacian on compact heisenberg manifolds |
| url | https://hdl.handle.net/1721.1/140343 |
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