• DOI:10.1007/s11040-022-09422-9
• Corpus ID: 236493143

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

@article{Ponge2021WeylsLA,
  title={Weyl’s Laws and Connes’ Integration Formulas for Matrix-Valued \$\$L\!\log \!L\$\$-Orlicz Potentials},
  author={Raphael Ponge},
  journal={Mathematical Physics, Analysis and Geometry},
  year={2021}
}

• Raphael Ponge
• Published 28 July 2021
• Mathematics
• Mathematical Physics, Analysis and Geometry

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.