  We present a semiclassical analysis of the quantum propagator of a particle
confined on one side by a steeply, monotonically rising potential. The models
studied in detail have potentials proportional to $x^{\alpha}$ for $x>0$; the
limit $\alpha\to\infty$ would reproduce a perfectly reflecting boundary, but at
present we concentrate on the cases $\alpha =1$ and 2, for which exact
solutions in terms of well known functions are available for comparison. We
classify the classical paths in this system by their qualitative nature and
calculate the contributions of the various classes to the leading-order
semiclassical approximation: For each classical path we find the action $S$,
the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of
interest because it gives an estimate of the error in the approximation and is
needed for computing higher-order approximations.) The resulting semiclassical
propagator can be used to rewrite the exact problem as a Volterra integral
equation, whose formal solution by iteration (Neumann series) is a
semiclassical, not perturbative, expansion. We thereby test, in the context of
a concrete problem, the validity of the two technical hypotheses in a previous
proof of the convergence of such a Neumann series in the more abstract setting
of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses
are violated when caustics develop in the classical dynamics; this opens up the
interesting future project of extending the methods to momentum space.
