  We discuss various issues associated with the calculation of the reduced
functional determinant of a special second order differential operator
$\boldmath${F}$ =-d^2/d\tau^2+\ddot g/g$, $\ddot g\equiv d^2g/d\tau^2$, with a
generic function $g(\tau)$, subject to periodic and Dirichlet boundary
conditions. These issues include the gauge-fixed path integral representation
of this determinant, the monodromy method of its calculation and the
combination of the heat kernel and zeta-function technique for the derivation
of its period dependence. Motivations for this particular problem, coming from
applications in quantum cosmology, are also briefly discussed. They include the
problem of microcanonical initial conditions in cosmology driven by a conformal
field theory, cosmological constant and cosmic microwave background problems.
