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