| dc.description.abstract | In the study of vacuum energy and the Casimir effect, it proves convenient to model the parallel conducting plates by a “soft” wall of the form v(z) = z^α rather than the standard Dirichlet wall. This model, for instance, does not violate the principle of virtual work under regularization, unlike the naive Dirichlet model. In previous research, the soft wall model was formalized for a massless scalar field, and expressions for the corresponding stress tensor were derived. Using these expressions, the energy density and pressure were calculated inside and out of the wall for the linear and quadratic walls, for which exact solutions exist. The limit of interest is α ≫ 1, which corresponds to the Dirichlet wall. Since a closed form expression for the Green function of the field equation cannot be found for α > 2, one must approximate it in order to use the previously derived expressions for the stress tensor. In this thesis, we conclude this research project. Using high order WKB and perturbation expansions of the associated Green function, we develop a robust approximation scheme in the regime where neither is valid. This approximation matches both expansions to an appropriate order in their domain of validity. We apply the developed scheme to the sextic soft wall and use it to compute the stress tensor inside the cavity for various conformal parameters. The consistency of the results is verified by checking known conservation laws and reproducing the energy density for the quadratic wall. To further verify our results, we compare the approximated stress tensor to a numerical counterpart, which is obtained by discretizing the separated field equation. To maximize accuracy and efficiency, we develop a customized numerical stiff linear boundary value solver which exploits key properties of the field equation. This solver is implemented in two different ways, which prioritize the concurrency of the solution process and the accuracy of the output. | en |
