  In this paper, we provide an approach for the calculation of one-loop
effective actions, vacuum energies, and spectral counting functions and discuss
the application of this approach in some physical problems. Concretely, we
construct the equations for these three quantities; this allows us to achieve
them by directly solving equations. In order to construct the equations, we
introduce shifted local one-loop effective actions, shifted local vacuum
energies, and local spectral counting functions. We solve the equations of
one-loop effective actions, vacuum energies, and spectral counting functions
for free massive scalar fields in $\mathbb{R}^{n}$, scalar fields in
three-dimensional hyperbolic space $H_{3}$ (the Euclidean Anti-de Sitter space
$AdS_{3}$), in $H_{3}/Z$ (the geometry of the Euclidean BTZ black hole), and in
$S^{1}$, and the Higgs model in a $(1+1)$-dimensional finite interval.
Moreover, in the above cases, we also calculate the spectra from the counting
functions. Besides exact solutions, we give a general discussion on approximate
solutions and construct the general series expansion for one-loop effective
actions, vacuum energies, and spectral counting functions. In doing this, we
encounter divergences. In order to remove the divergences, renormalization
procedures are used. In this approach, these three physical quantities are
regarded as spectral functions in the spectral problem.

High Energy Physics - Theory