  We discuss a family of time-inhomogeneous two-dimensional diffusions, defined
over a finite time interval $[0,T]$, having transition density functions that
are expressible in terms of the integral kernels for negative exponentials of
the two-dimensional Schr\"odinger operator with a point potential at the
origin. These diffusions have a singular drift pointing in the direction of the
origin that is strong enough to enable the possibly of visiting there, in
contrast to a two-dimensional Brownian motion. Our main focus is on
characterizing a local time process at the origin analogous to that for a
one-dimensional Brownian motion and on studying the law of its process inverse.
