Casimir energy between two concentric half spheres

We consider the Casimir effect for the massless scalar field between two concentric half spheres. The Casimir energy of a massless scalar field subject to the boundary conditions between two concentric half spheres are calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We devoleped a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions without the use of the asymptotic Debye approximation. All contributions in Casimir energy come from the high frequency modes at fixed ℓ between two surface boundary conditions. We have obtained the negative sign of the Casimir energy for a massless scalar field between two concentric half spheres. The massless scalar field is regarded as interesting both by itself and as the key to describing of the electromagnetic case.