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Casimir effect between two spheres at small separations

We consider the Casimir interaction between two spheres at zero and finite temperature, for both scalar fields and electromagnetic fields. Of particular interest is the asymptotic expansions of the Casimir free energy when the distance between the spheres is small. The scenario where one sphere is inside the other is discussed in detail. At zero temperature, we compute analytically the leading and the next-to-leading order terms from the functional determinant representation of the Casimir energy. As expected, the leading order term agrees with the proximity force approximation. The results for the next-to-leading order terms are new. In the limit where the radius of the outer sphere goes to infinity, the results for the sphere-plane geometry are reproduced. At finite temperature, the leading order term is computed and it is found to agree completely with the proximity force approximation in the medium and high temperature regions. For the scenario where two spheres are outside each other, analogous results are obtained. In the case of Dirichlet boundary conditions on both spheres, the next-to-leading order term of the zero temperature Casimir energy is found to agree with that computed recently using derivative expansion.