  We prove an analogue of Chernoff's theorem for the Laplacian $
\Delta_{\mathbb{H}} $ on the Heisenberg group $ \mathbb{H}^n.$ As an
application, we prove Ingham type theorems for the group Fourier transform on $
\mathbb{H}^n $ and also for the spectral projections associated to the
sublaplacian.
