Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger\n Operators on Curved Noncommutative Tori

2021/03/31 by Edward McDonald, McDonald, Edward, Raphaël Ponge +1
Mathematics · #47B10 #58B34 #81Q10 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2103.16869

openalex publication_date 2021/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a previous paper we established Cwikel-type estimates on noncommutative\ntori and used them to get analogues in this setting of the\nCwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative\neigenvalues of fractional Schr "odinger operators. In this paper, we focus on\n"curved" NC tori, where the role of the usual Laplacian is played by\nLaplace-Beltrami operators associated with arbitrary Riemannian metrics. The\nCwikel-type estimates of our previous paper are extended to pseudodifferential\noperators and powers of Laplace-Beltrami operators. There are several\napplications of these estimates. First, we get L_p-versions of the usual\nformula for the trace of psidos on NC tori, i.e., for combinations of\n psidos with L_p-position operators. Next, we get L_p-versions of the\nanalogues for NC tori Connes' trace theorem and Connes' integration formula.\nThey give formulas for the NC integrals (a.k.a. Dixmier traces) of products of\nL_p-position operators with psidos or powers of the Laplace-Beltrami\noperators. Moreover, by combining our Cwikel-type estimates with suitable\nversions of the Birman-Schwinger principle we get versions of the CLR and\nLieb-Thirring inequalities for negative eigenvalues of fractional Schr "odinger\noperators associated with powers of Laplace-Beltrami operators and\nL_p-potentials. As in the original Euclidean case the Lieb-Thirring\ninequalities imply a dual Sobolev inequality for orthonormal families. Finally,\nwe discuss spectral asymptotics and semiclassical Weyl's laws for the our\nclasses of operators on curved NC tori. This superseded a previous conjecture\nin our previous paper.\n

Citations

• The Gauss-Bonnet Theorem for the noncommutative two torus
• Cwikel's bound reloaded
• Local invariants of non-commutative tori
• Cwikel-Solomyak estimates on tori and Euclidean spaces
• Connes Integration Formula without singular traces

Related

• Trace identities for commutators, with applications to the distribution of eigenvalues
• Cwikel's theorem and the CLR inequality
• Trace Formulas in Noncommutative Geometry
• Spectral asymptotics of all the eigenvalues of Schr "odinger operators\n on flat tori
• Cwikel-Lieb-Rozenblum type inequalities for Hardy-Schrödinger operator