A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds

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.