  We derive an analytical expression for the transition path time (TPT)
distribution for a one-dimensional particle crossing a parabolic barrier. The
solution is expressed in terms of the eigenfunctions and eigenvalues of the
associated Fokker-Planck equation. The particle performs an anomalous dynamics
generated by a power-law memory kernel, which includes memoryless Markovian
dynamics as a limiting case. Our result takes into account absorbing boundary
conditions, extending existing results obtained for free boundaries. We show
that TPT distributions obtained from numerical simulations are in excellent
agreement with analytical results, while the typically employed free boundary
conditions lead to a systematic overestimation of the barrier height. These
findings may be useful in the analysis of experimental results on transition
path times. A web tool to perform this analysis is freely available.
