Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials

TL;DR

This paper extends Weyl's laws and Connes' integration formulas to matrix-valued $L ext{log}L$-Orlicz potentials, providing new semiclassical and spectral asymptotics for critical Schrödinger operators.

Contribution

It establishes Weyl's laws and Connes' integration formulas for matrix-valued $L ext{log}L$-Orlicz potentials on manifolds, generalizing previous results to a broader class of potentials.

Findings

Derived CLR-type inequalities for matrix-valued potentials.

Proved semiclassical Weyl's laws for critical Schrödinger operators.

Extended Connes' integration formula to matrix-valued $L ext{log}L$-Orlicz potentials.

Abstract

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.