  A $d$-dimensional quantum model system confined to a general hypercubical
geometry with linear spatial size $L$ and ``temporal size'' $1/T $ ($T$ -
temperature of the system) is considered in the spherical approximation under
periodic boundary conditions. For a film geometry in different space dimensions
$\frac 12\sigma <d<\frac 32\sigma $, where $0<\sigma \leq 2$ is a parameter
controlling the decay of the long-range interaction, the free energy and the
Casimir amplitudes are given. We have proven that, if $d=\sigma$, the Casimir
amplitude of the model, characterizing the leading temperature corrections to
its ground state, is $\Delta =-16\zeta(3)/[5\sigma(4\pi)^{\sigma/2}\Gamma
(\sigma /2)]$. The last implies that the universal constant $\tilde{c}=4/5$ of
the model remains the same for both short, as well as long-range interactions,
if one takes the normalization factor for the Gaussian model to be such that
$\tilde{c}=1$. This is a generalization to the case of long-range interaction
of the well known result due to Sachdev. That constant differs from the
corresponding one characterizing the leading finite-size corrections at zero
temperature which for $d=\sigma=1$ is $\tilde c=0.606$.
