  In slow collisions of two bare nuclei with the total charge number larger
than the critical value, $Z_{\rm cr} \approx 173$, the initially neutral vacuum
can spontaneously decay into the charged vacuum and two positrons. Detection of
the spontaneous emission of positrons would be the direct evidence of this
fundamental phenomenon. However, the spontaneous emission is generally masked
by the dynamical positron emission, which is induced by a strong time-dependent
electric field created by the colliding nuclei. In our recent paper [I.A.
Maltsev et al., Phys. Rev. Lett. 123, 113401 (2019)] it has been shown that the
spontaneous pair production can be observed via measurements of the
pair-production probabilities for a given set of nuclear trajectories. In the
present paper, we have significantly advanced this study by exploring
additional aspects of the process we are interested in. We calculate the
positron energy spectra and find that these spectra can give a clear signature
of the transition from the subcritical to the supercritical regime. It is found
that focusing on a part of the positron spectrum, which accounts for the energy
region where the spontaneously created positrons can contribute, allows to get
a much stronger evidence of the transition to the supercritical mode, making it
very well pronounced in collisions, for example, of two uranium nuclei. The
possibility of extending this study to collisions of bare nuclei with neutral
atoms is also considered. The probability of a vacancy in the lowest-energy
state of a quasimolecule which is formed in collisions of a bare U nucleus with
neutral U and Cm atoms has been calculated. The relatively large values of this
probability make such collisions suitable for observing the vacuum decay.
