Vacuum energy density and pressure inside a soft wall

Abstract: 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.