Investigation of continuous-time quantum walks via spectral analysis and Laplace transform

Continuous-time quantum walk (CTQW) on a given graph is investigated by using the techniques of the spectral analysis and inverse Laplace transform of the Stieltjes function (Stieltjes transform of the spectral distribution) associated with the graph. It is shown that, the probability amplitude of observing the CTQW at a given site at time $t$ is related to the inverse Laplace transformation of the Stieltjes function, namely, one can calculate the probability amplitudes only by taking the inverse laplace transform of the function $iG_{\mu}(is)$, where $G_{\mu}(x)$ is the Stieltjes function of the graph. The preference of this procedure is that, there is no any need to know the spectrum of the graph