Weyl’s Laws and Connes’ Integration Formulas for Matrix-Valued \(L\!\log \!L\)-Orlicz Potentials

Thanks to the Birman-Schwinger principle, Weyl’s laws for Birman-Schwinger operators yields semiclassical Weyl’s laws for the corresponding Schrödinger 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, we show that, 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 Schrödinger 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.