  In the study of quantum vacuum energy and the Casimir effect, it is desirable
to model the conductor by a potential of the form $V(z)=z^\alpha$. This "soft
wall" model was proposed so as to avoid the violation of the principle of
virtual work under ultraviolet regularization that occurs for the standard
Dirichlet wall. The model was formalized for a massless scalar field, and the
expectation value of the stress tensor has been expressed in terms of the
reduced Green function of the equation of motion. In the limit of interest,
$\alpha \gg 1$, which approximates a Dirichlet wall, a closed-form expression
for the reduced Green function cannot be found, so piecewise approximations
incorporating the perturbative and WKB expansions of the Green function, along
with interpolating splines in the region where neither expansion is valid, have
been developed. After reviewing this program, in this article we apply the
scheme to the wall with $\alpha=6$ and use it to compute the renormalized
energy density and pressure inside the cavity for various values of the
conformal parameter. The consistency of the results is verified by comparison
to their numerical counterparts and verification of the trace anomaly and the
conservation law. Finally, we use the approximation scheme to reproduce the
energy density inside the quadratic wall, which was previously calculated
exactly but with some uncertainty.
