Perturbative check on the Casimir energies of nondispersive dielectric spheres

For an optically dilute solid sphere of radius cc and dielectric constant e psilon independent of frequency, the Casimir energy Delta is evaluated to s econd order in gamma = (1 - 1/epsilon), subject to an exponential cut-off 1 /lambda on wavenumbers, using only standard perturbation theory and element ary mathematics. It is hoped that this can serve to elucidate other far mor e elaborate methods that aim to determine Delta exactly by summing zero-poi nt energies. For the electromagnetic field, the perturbative result reads Delta(em) = -gamma 3/2 pi(2) V/lambda(4) + gamma(2) { - 3/128 pi(2) V/lambd a(4) + 7/360 pi(3) S/lambda(3) - 1/20 pi(2) 1/lambda + 23/1536 pi 1/a} +... with V the volume and S the surface area. The term of order gamma(2) is rel ated in a simple way to the Casimir-Polder (retarded) potential between pol arizable bodies. This relation also yields some insight into the net pressu re on a thin spherical shell.