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Eigenvalues of singular measures and Connes noncommutative integration
TL;DR: In this paper, the authors considered the case of a singular measure and a pseudodifferential operator and established eigenvalue asymptotics of these operators for a class of measures, including those supported on uniformly rectifiable sets.
Abstract: For a singular measure $\mu$, Ahlfors regular of order $\alpha>0,$ with compact support in $\mathbb{R}^{\mathbf{N}}$ and a pseudodifferential operator $\mathbf{A}$ of order $-l=-\mathbf{N}/2$ we consider the compact operator $\mathbf{T}(P,\mathbf{A}) = \mathbf{A}^*P\mathbf{A}.$ Here $P$ is the signed measure, $P=V\mu$ with density $V$ belonging to the Orlicz class $L^{\Psi,\mu}$ with $\Psi(t)=(t+1)\log(t+1)-t.$ Using eigenvalue estimates for such operators, obtained in \texttt{arXiv:2011.14877}, we establish eigenvalue asymptotics of $\mathbf{T}(P,\mathbf{A})$ for a class of measures, including the ones supported on uniformly rectifiable sets. These results lead to the measurability in the sense of A.Connes of operators $\mathbf{T}(P,\mathbf{A})$ and a formula for the singular trace of these operators, producing a noncommutative version of integral with respect to singular measure.
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TL;DR: In this article, Cwikel-type estimates and the CLR inequality for non-commutative tori were established for pseudodifferential operators and curved non-complementary tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary densities.
Abstract: In a previous paper we established Cwikel-type estimates and the CLR inequality for noncommutative tori In this follow-up paper we extend these results to pseudodifferential operators and to curved noncommutative tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary densities and Riemannian metrics The Cwikel estimates are used to get several $L_2$ and $L_{1^+}$ Dixmier trace formulas Here $L_{1^+}$ is meant as the intersection of all $L_p$-spaces with $p>1$ This extends previous Dixmier trace formulas to the $L_2$ and $L_{1^+}$ settings Combining the Cwikel estimates with a borderline version of the Birman-Schwinger principle leads to a CLR-type inequality for the number of negative eigenvalues of (fractional) Schrodinger operators that are built out of fractional Laplace-Beltrami operators and $L_p$-potentials We also conjecture a semi-classical Weyl's law for such operators
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TL;DR: In this article, the authors prove a Tauberian theorem for singular values of noncommuting operators which allows them to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality.
Abstract: We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes' notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(\mathcal{A},H,D)$ and self-adjoint $V \in \mathcal{A}$, there holds \[\lim_{h\downarrow 0} h^p\mathrm{Tr}(\chi_{(-\infty,0)}(h^2D^2+V)) = \int V_-^{\frac{p}{2}}|ds|^p.\] where $\int$ is Connes' noncommutative integral.
3 citations
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TL;DR: In this paper, the Weyl's laws for critical Schrodinger operators associated with matrix-valued $L\log L$-Orlicz potentials were shown to imply a strong version of Connes' integration formula.
Abstract: Thanks to the Birman-Schwinger principle, Weyl's laws for Birman-Schwinger operators yields semiclassical Weyl's laws for the corresponding Schrodinger 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 Schrodinger 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.
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TL;DR: In this paper, the authors considered compact, Birman-Schwinger type pseudodifferential operators with singular Borel measures and proved a proper version of H.Weyl's asymptotic law for eigenvalues with order depending on dimensional characteristics of the measure.
Abstract: In a domain $\Omega\subseteq \mathbb{R}^\mathbf{N}$ we consider compact, Birman-Schwinger type, operators of the form $\mathbf{T}_{P,\mathfrak{A}}=\mathfrak{A}^*P\mathfrak{A}$; here $P$ is a singular Borel measure in $\Omega$ and $\mathfrak{A}$ is a noncritical order $-l
e -\mathbf{N}/2$ pseudodifferential operator. For a class of such operators, we obtain estimates and a proper version of H.Weyl's asymptotic law for eigenvalues, with order depending on dimensional characteristics of the measure. A version of the CLR estimate for singular measures is proved. For non-selfadjoint operators of the form $P_2 \mathfrak{A} P_1$ and $\mathfrak{A}_2 P \mathfrak{A}_1$ with singular measures $P,P_1,P_2$ and negative order pseudodifferential operators $\mathfrak{A},\mathfrak{A}_1,\mathfrak{A}_2$ we obtain estimates for singular numbers.
2 citations
TL;DR: In this article , the authors established Lieb-Thirring type estimates for the Schrödinger operator with a singular measure serving as potential, and proved that these estimates are correct.
Abstract: Abstract We establish Lieb–Thirring type estimates for the Schrödinger operator with a singular measure serving as potential.
2 citations
References
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01 Jun 1969
TL;DR: In this article, Grassmann algebras of a vectorspace have been studied in the context of the calculus of variations, and a glossary of some standard notations has been provided.
Abstract: Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 Simple m-vectors 1.8 Mass and comass 1.9 The symmetric algebra of a vectorspace 1.10 Symmetric forms and polynomial functions Chapter 2 General measure theory 2.1 Measures and measurable sets 2.2 Borrel and Suslin sets 2.3 Measurable functions 2.4 Lebesgue integrations 2.5 Linear functionals 2.6 Product measures 2.7 Invariant measures 2.8 Covering theorems 2.9 Derivates 2.10 Caratheodory's construction Chapter 3 Rectifiability 3.1 Differentials and tangents 3.2 Area and coarea of Lipschitzian maps 3.3 Structure theory 3.4 Some properties of highly differentiable functions Chapter 4 Homological integration theory 4.1 Differential forms and currents 4.2 Deformations and compactness 4.3 Slicing 4.4 Homology groups 4.5 Normal currents of dimension n in R(-63) superscript n Chapter 5 Applications to the calculus of variations 5.1 Integrands and minimizing currents 5.2 Regularity of solutions of certain differential equations 5.3 Excess and smoothness 5.4 Further results on area minimizing currents Bibliography Glossary of some standard notations List of basic notations defined in the text Index
6,513 citations
Book•
01 Jan 1966
TL;DR: The merite as discussed by the authors is a date marque une date dans le progres des mathematiques and de la physique en levant l'ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens and l'inacceptabilite de leurs formules au regard de la rigueur mathematiques.
Abstract: Ce traite a marque une date dans le progres des mathematiques et de la physique en levant l’ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens et l’inacceptabilite de leurs formules au regard de la rigueur mathematiques Le merite revient a Laurent Schwartz d’avoir englobe dans une theorie qui est a la fois une synthese et une simplifications, des procedes heterogenes et souvent incorrects utilises dans des domaines tres divers
Une definition correcte et une etude systematique de ces etres nouveaux, les distributions, leur ont donne droit de cite dans l’usage courant Leur utilisation extensive dans de nombreuses branches des mathematiques pures et appliquees, de la physique et des sciences de l’ingenieur fait de ce livre un classique des mathematiques modernes
4,197 citations
Book•
01 Jan 1991
TL;DR: A reference/text for mathematicians or students involved in analysis, differential equations, probability theory, and the study of integral operators where only Lebesgue spaces were used in the past is.
Abstract: A reference/text for mathematicians or students involved in analysis, differential equations, probability theory, and the study of integral operators where only Lebesgue spaces were used in the past. Updates and extends the pioneering work by Krasnosel'skii and Rutickii in their 1958 treatise on Orl
1,948 citations
Book•
01 Jan 1985TL;DR: In this paper, a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension is presented, including questions of local density and the existence of tangents of such sets, and the dimensional properties of their projections in various directions.
Abstract: This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.
1,802 citations