We present a new spectral-theoretic derivation of the density of states \( \rho \left( \lambda \right) \) for the sublaplacian operator on the Heisenberg group \(\mathbb {H}^{n}\) . Our method exploits a fundamental connection between this operator and the magnetic Laplacian operator in \( \mathbb {C}^{n}\) , linked via a Fourier transform. While the resolvent kernel for the sublaplacian operator is established from a prior work (and whose consistency with Folland’s fundamental solution is a key validation of its form), our core contribution lies in the direct application of the associated spectral density kernel \(dE_{\lambda }/d\lambda \) to obtain \(\rho \left( \lambda \right) =\gamma _{n}\lambda ^{n}, \gamma _{n}>0\) is a constant. This approach provides an independent, \(L{{}^2}\) -spectral alternative to the harmonic analysis techniques used by Strichartz (J. Fourier Anal. Appl. 18, 626–659, 2012) to find the integrated density of states, whose derivative confirms our result. Additionally, we provide a general formula for the integrated density of states of the magnetic Laplacian operator in \(\mathbb {C}^{n}\) , extending Nakamura’s one-dimensional result (Nakamura, J. Funct. Anal. 179(1), 136–152, 2001). This work highlights the practical utility of the connection between these two operators, demonstrating that the spectral theory of the Heisenberg sublaplacian operator can be effectively advanced by transferring results from the well-studied context of magnetic Hamiltonians in \(\mathbb {C} ^{n}\) .
