Transition from discrete to continuous time of arrival distribution for a quantum particle

We show that the Kijowski distribution for time of arrivals in the entire real line is the limiting distribution of the time of arrival distribution in a confining box as its length increases to infinity. The dynamics of the confined time of arrival eigenfunctions is also numerically investigated and demonstrated that the eigenfunctions evolve to have point supports at the arrival point at their respective eigenvalues in the limit of arbitrarilly large confining lengths, giving insight into the ideal physical content of the Kijowsky distribution.