  We introduce a unified framework for solving first passage times of
time-homogeneous diffusion processes. According to the killed version potential
theory and the perturbation theory, we are able to deduce closed-form solutions
for probability densities of single-sided level crossing problem. The framework
is applicable to diffusion processes with continuous drift functions, and a
recursive system in the frequency domain has been provided. Besides, we derive
a probabilistic representation for error estimation. The representation can be
used to evaluate deviations in perturbed density functions. In the present
paper, we apply the framework to Ornstein-Uhlenbeck and Bessel processes to
find closed-form approximations for their first passage times; another
successful application is given by the exponential-Shiryaev process. Numerical
results are provided at the end of this paper.
