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We compute the generic mode sum that quantifies the effect on the spectrum ofa harmonic field when a spherical shell is inserted into vacuum. Thisencompasses a variety of problems including the Weyl spectral problem and theCasimir effect of quantum electrodynamics. This allows us to resolve severallong-standing controversies regarding the question of universality of theCasimir self-energy; the resolution comes naturally through the connection tothe Weyl problem. Specifically we demonstrate that in the case of a scalarfield obeying Dirichlet or Neumann boundary conditions on the shell surface theCasimir self-energy is cutoff-dependent while in the case of theelectromagnetic field perturbed by a conductive shell the Casimir self-energyis universal. We additionally show that an analog non-relativistic Casimireffect due to zero-point magnons takes place when a non-magnetic sphericalshell is inserted inside a bulk ferromagnet.

Weyl problem and Casimir effects in spherical shell geometry