  Thanks to the Birman-Schwinger principle, Weyl's laws for Birman-Schwinger
operators yields semiclassical Weyl's laws for the corresponding Schr\"odinger
operators. In a recent preprint Rozenblum established quite general Weyl's laws
for Birman-Schwinger operators associated with pseudodifferential operators of
critical order and potentials that are product of $L\log L$-Orlicz functions
and Alfhors-regular measures supported on a submanifold. In this paper, for
matrix-valued $L\log L$-Orlicz potentials supported on the whole manifold,
Rozenblum's results are direct consequences of the Cwikel-type estimates on
tori recently established by Sukochev-Zanin. As applications we obtain CLR-type
inequalities and semiclassical Weyl's laws for critical Schr\"odinger operators
associated with matrix-valued$L\log L$-Orlicz potentials. Finally, we explain
how the Weyl's laws of this paper imply a strong version of Connes' integration
formula for matrix-valued $L\log L$-Orlicz potentials.
