Casimir effect for soft boundaries

Abstract

In quantum field theory with confining ``hard'' (e.g., Dirichlet) boundaries, the latter are represented in the Schr\"odinger equation defining spatial quantum modes by infinite step-function potentials. One can instead introduce confining ``soft'' boundaries, represented in the mode equation by some smoothly increasing potential function. Here the global Casimir energy is calculated for a scalar field confined by harmonic-oscillator (HO) potentials in 1, 2, and 3 dimensions. Combinations of HO and Dirichlet boundaries are also considered. Some results differ in sign from comparable hard-wall ones.