It is shown that if $Z_3$ (where $Z_3$ is the photon wave-function renormalization constant) is assumed finite and the nonasymptotic part $h$ of the renormalized photon propagator vanishes with power-law behavior, then all the remaining renormalization constants in scalar electrodynamics can be made finite order by order, except the charged-meson self-mass $\delta m^2$. The condition that $\delta m^2$ be finite forces the asymptotic coupling ${\alpha }_0$ to satisfy at least one eigenvalue equation. A second eigenvalue condition for ${\alpha }_0$ emerges from the requirement that the theory have a Hermitian Lagrangian. Finally, on the basis of the renormalization group, we expect that the initial assumption of a finite value of $Z_3$ is self-consistent only if ${\alpha }_0$ satisfies a third eigenvalue condition. Hence, we conjecture that a completely finite, closed theory of scalar electrodynamics is probably internally inconsistent. Assuming that $h$ falls off sufficiently rapidly, we are able to show that the meson propagator has a very simple asymptotic form for momenta much greater than its physical mass.

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