  We consider a one-dimensional diffusion which solves a stochastic
differential equation with Borel-measurable coefficients in an open interval.
We allow for the endpoints to be inaccessible or absorbing. Given a
Borel-measurable function $r$ that is uniformly bounded away from 0, we
establish a new analytic representation of the $r$-potential of a continuous
additive functional of the diffusion. We also characterize the value function
of an optimal stopping problem with general reward function as the unique
solution of a variational inequality (in the sense of distributions) with
appropriate growth or boundary conditions. Furthermore, we establish several
other characterisations of the solution to the optimal stopping problem,
including a generalisation of the so-called "principle of smooth fit".

Probability