  We present exact results for the spectrum of the fractional Laplacian in a
bounded domain and apply them to First Passage Time (FPT) Statistics of L\'evy
flights. We specifically show that the average is insufficient to describe the
distribution of FPT, although it is the only quantity available in the existing
literature. In particular, we show that the FPT distribution is not peaked
around the average, and that knowledge of the whole distribution is necessary
to describe this phenomenon. For this purpose, we provide an efficient method
to calculate higher order cumulants and the whole distribution.
