  Basic quantum processes (such as particle creation, reflection, and
transmission on the corresponding Klein steps) caused by inverse-square
electric fields are calculated. These results represent a new example of exact
nonperturbative calculations in the framework of QED. The inverse-square
electric field is time-independent, inhomogeneous in the $x$-direction, and is
inversely proportional to $x$ squared. We find exact solutions of the Dirac and
Klein-Gordon equations with such a field and construct corresponding in- and
out-states. With the help of these states and using the techniques developed in
the framework of QED with $x$-electric potential steps, we calculate
characteristics of the vacuum instability, such as differential and total mean
numbers of particles created from the vacuum and vacuum-to-vacuum transition
probabilities. We study the vacuum instability for two particular backgrounds:
for fields widely stretches over the $x$-axis (small-gradient configuration)
and for the fields sharply concentrates near the origin $x=0$ (sharp-gradient
configuration). We compare exact results with ones calculated numerically.
Finally, we consider the electric field configuration, composed by
inverse-square fields and by an $x$-independent electric field between them to
study the role of growing and decaying processes in the vacuum instability.
