Abstract. We study the Casimir effect for scalar fields with general curvature coupling subject to mixed boundary conditions (1 + β m n µ ∂ µ )ϕ = 0 at x = a m on one (m = 1) and two (m = 1, 2) parallel plates at a distance a ≡ a 2 − a 1 from each other. Making use of the generalized AbelPlana formula previously established by one of the authors [1], the Casimir energy densities are obtained as functions of β 1 and of β 1 ,β 2 ,a, respectively. In the case of two parallel plates, a decomposition of the total Casimir energy into volumic and superficial contributions is provided. The possibility of finding a vanishing energy for particular parameter choices is shown, and the existence of a minimum to the surface part is also observed. We show that there is a region in the space of parameters defining the boundary conditions in which the Casimir forces are repulsive for small distances and attractive for large distances. This yields to an interesting possibility for stabilizing the distance between the plates by using the vacuum forces.

The vacuum expectation values for the energy-momentum tensor of a massive scalar field with general curvature coupling and obeying the Robin boundary condition on spherically symmetric boundaries in D-dimensional space are investigated. The expressions are derived for the regularized vacuum energy density and radial and azimuthal stress components (i) inside and outside a single spherical surface and (ii) in the intermediate region between two concentric spheres. A regularization procedure is carried out by making use of the generalized Abel-Plana formula for the series over zeros of cylinder functions. The asymptotic behavior of the vacuum densities near the sphere and at large distances is investigated. A decomposition of the Casimir energy into volumic and surface parts is provided for both cases (i) and (ii). We show that the mode sum energy, evaluated as a sum of the zero-point energy for each normal mode of frequency, and the volume integral of the energy density in general are different, and argue that this difference is due to the existence of an additional surface energy contribution.

We evaluate the Casimir energy and force for a massive scalar field with general curvature coupling parameter, subject to Robin boundary conditions on two codimension-one parallel plates, located on a (D + 1)-dimensional background spacetime with an arbitrary internal space. The most general case of different Robin coefficients on the two separate plates is considered. With independence of the geometry of the internal space, the Casimir forces are seen to be attractive for special cases of Dirichlet or Neumann boundary conditions on both plates and repulsive for Dirichlet boundary conditions on one plate and Neumann boundary conditions on the other. For Robin boundary conditions, the Casimir forces can be either attractive or repulsive, depending on the Robin coefficients and the separation between the plates, what is actually remarkable and useful. Indeed, we demonstrate the existence of an equilibrium point for the interplate distance, which is stabilized due to the Casimir force, and show that stability is enhanced by the presence of the extra dimensions. Applications of these properties in braneworld models are discussed. Finally, the corresponding results are generalized to the geometry of a piston of arbitrary cross section.

We investigate the vacuum expectation value of the fermionic current induced by a magnetic flux in a (2 þ 1)-dimensional conical spacetime in the presence of a circular boundary. On the boundary the fermionic field obeys the MIT bag boundary condition. For irregular modes, a special case of boundary conditions at the cone apex is considered, when the MIT bag boundary condition is imposed at a finite radius, which is then taken to zero. We observe that the vacuum expectation values for both the charge density and azimuthal current are periodic functions of the magnetic flux with the period equal to the flux quantum whereas the expectation value of the radial component vanishes. For both exterior and interior regions, the expectation values of the current are decomposed into boundary-free and boundary-induced parts. For a massless field the boundary-free part in the vacuum expectation value of the charge density vanishes, whereas the presence of the boundary induces nonzero charge density. Two integral representations are given for the boundary-free part in the case of a massive fermionic field for arbitrary values of the opening angle of the cone and magnetic flux. The behavior of the induced fermionic current is investigated in various asymptotic regions of the parameters. At distances from the boundary larger than the Compton wavelength of the fermion particle, the vacuum expectation values decay exponentially with the decay rate depending on the opening angle of the cone. We make a comparison with the results already known from the literature for some particular cases.