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

Abstract

Thanks to the Birman-Schwinger principle, Weyl's laws for Birman-Schwinger operators yields semiclassical Weyl's laws for the corresponding 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, 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.

23 pages. arXiv admin note: text overlap with arXiv:2107.01242

References in corpus (5)

• Traces of compact operators and the noncommutative residue
• Cwikel-Solomyak estimates on tori and Euclidean spaces
• Connes Integration Formula without singular traces
• Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry
• Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori