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

Abstract

Let \(M= \Gamma \setminus \mathbb {H}_d\) be a compact quotient of the d-dimensional Heisenberg group \(\mathbb {H}_d\) by a lattice subgroup \(\Gamma \). We show that the eigenvalue counting function \(N^\alpha \left( \lambda \right) \) for any fixed element of a family of second order differential operators \(\left\{ \mathcal {L}_\alpha \right\} \) on M has asymptotic behavior \(N^\alpha \left( \lambda \right) \sim C_{d,\alpha } {\text {vol}}\left( M\right) \lambda ^{d + 1}\), where \(C_{d,\alpha }\) is a constant that only depends on the dimension d and the parameter \(\alpha \). 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 \({\mathcal {L}}_{\alpha }\) and Karamata’s Tauberian theorem.