We describe the analytic continuation of the heat kernel on the Heisenberg group $\Hn(\R)$. As a consequence, we show that the convolution kernel corresponding to the Schr\"odinger operator $e^{isL}$ is a smooth function on $\Hn(\R) \setminus S_s$, where $S_s=\{(0,0,\pm sk)\in\Hn(\R): k = n, n+2, n+4, \dots \}$. At every point of $S_s$ the convolution kernel of $e^{isL}$ has a singularity of Calder\'on--Zygmund type.