  The superfluid density of overdoped La$_{2-x}$Sr$_x$CuO$_4$ thin films of
high quality have been measured with Tc (defined by the onset of the Meissner
effect) from 5.1 to 41.6 K by Bosovic et al. Given this Tc the superfluid
density shows no clear evidence of critical fluctuations and no indication of
vortex unbinding as T->Tc. Nevertheless, the superfluid density displays the
expected behavior for for a quantum phase transition (QPT) in the (3+1)D-xy
universality class, rho(T=0)~Tc^2. However, this relation is also a hallmark of
dirty superconductors, treated in the mean-field approximation. Here we attempt
to clear out the nature of the suppression of ro(T) as Tc->0. Noting that for
any finite system the continuous transition will be rounded we perform a finite
size scaling analysis. It uncovers that the ro(T) data data are consistent with
a finite length limited 3D-xy transition. In some films it is their thickness
and in others their inhomogeneity that determines the limiting length. Having
established the precondition for the occurrence of a QPT mapping on the (3+1)
D-xy model, we explore the consistency with the hallmarks of this transition.
In particular with the relations rho(T)/rho(0)=1-ycT/Tc, rho((0)^Tc^2,
yc=alpha*Tc/ro(0), where alpha is the coefficient in rho(T)=rho(0)-alphaT as
T->0. The emerging agreement with these characteristics points clearly to a
quantum fluctuations induced suppression, revealing the crossover from the
thermal to the quantum critical regime as Tc ->0. In the classical-quantum
mapping it corresponds to a 3D to (3 + 1)D crossover.
