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Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials
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 article , Cwikel-type estimates for non-commutative tori for any dimension n ≥ 2 were derived, and they were used to derive the Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators.
Abstract: In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive Cwikel–Lieb–Rozenblum inequalities and Lieb–Thirring inequalities for negative eigenvalues of fractional Schrödinger operators on noncommutative tori. The latter leads to a Sobolev inequality for noncommutative tori. On the way, we establish a “borderline version” of the abstract Birman–Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.
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TL;DR: In this article , the authors proposed a method to solve the problem of the lack of information in the context of data sharing in the field of software engineering. But they did not specify how to do it.
Abstract: Получен аналог формулы интегрирования Конна, который дает конкретную асимптотику собственных значений. Это существенно расширяет класс функций на компактных римановых многообразиях, интегрируемых в некоммутативном смысле.
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Book•
01 Jan 1979TL;DR: In this paper, Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for trace, determinant, and Lidskii's theorem are discussed.
Abstract: Preliminaries Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for $\mathcal J_P$ Trace, determinant, and Lidskii's theorem $f(x)g(-i
abla)$ Fredholm theory Scattering with a trace condition Bound state problems Lots of inequalities Regularized determinants and renormalization in quantum field theory An introduction to the theory on a Banach space Borel transforms, the Krein spectral shift, and all that Spectral theory of rank one perturbations Localization in the Anderson model following Aizenman-Molchanov The Xi function Addenda Bibliography Index.
2,465 citations
Book•
01 Apr 1992
TL;DR: In this article, the authors present a formal solution for the trace of the heat kernel on Euclidean space, and show that the trace can be used to construct a heat kernel of an equivariant vector bundle.
Abstract: 1 Background on Differential Geometry.- 1.1 Fibre Bundles and Connections.- 1.2 Riemannian Manifolds.- 1.3 Superspaces.- 1.4 Superconnections.- 1.5 Characteristic Classes.- 1.6 The Euler and Thorn Classes.- 2 Asymptotic Expansion of the Heat Kernel.- 2.1 Differential Operators.- 2.2 The Heat Kernel on Euclidean Space.- 2.3 Heat Kernels.- 2.4 Construction of the Heat Kernel.- 2.5 The Formal Solution.- 2.6 The Trace of the Heat Kernel.- 2.7 Heat Kernels Depending on a Parameter.- 3 Clifford Modules and Dirac Operators.- 3.1 The Clifford Algebra.- 3.2 Spinors.- 3.3 Dirac Operators.- 3.4 Index of Dirac Operators.- 3.5 The Lichnerowicz Formula.- 3.6 Some Examples of Clifford Modules.- 4 Index Density of Dirac Operators.- 4.1 The Local Index Theorem.- 4.2 Mehler's Formula.- 4.3 Calculation of the Index Density.- 5 The Exponential Map and the Index Density.- 5.1 Jacobian of the Exponential Map on Principal Bundles.- 5.2 The Heat Kernel of a Principal Bundle.- 5.3 Calculus with Grassmann and Clifford Variables.- 5.4 The Index of Dirac Operators.- 6 The Equivariant Index Theorem.- 6.1 The Equivariant Index of Dirac Operators.- 6.2 The Atiyah-Bott Fixed Point Formula.- 6.3 Asymptotic Expansion of the Equivariant Heat Kernel.- 6.4 The Local Equivariant Index Theorem.- 6.5 Geodesic Distance on a Principal Bundle.- 6.6 The heat kernel of an equivariant vector bundle.- 6.7 Proof of Proposition 6.13.- 7 Equivariant Differential Forms.- 7.1 Equivariant Characteristic Classes.- 7.2 The Localization Formula.- 7.3 Bott's Formulas for Characteristic Numbers.- 7.4 Exact Stationary Phase Approximation.- 7.5 The Fourier Transform of Coadjoint Orbits.- 7.6 Equivariant Cohomology and Families.- 7.7 The Bott Class.- 8 The Kirillov Formula for the Equivariant Index.- 8.1 The Kirillov Formula.- 8.2 The Weyl and Kirillov Character Formulas.- 8.3 The Heat Kernel Proof of the Kirillov Formula.- 9 The Index Bundle.- 9.1 The Index Bundle in Finite Dimensions.- 9.2 The Index Bundle of a Family of Dirac Operators.- 9.3 The Chern Character of the Index Bundle.- 9.4 The Equivariant Index and the Index Bundle.- 9.5 The Case of Varying Dimension.- 9.6 The Zeta-Function of a Laplacian.- 9.7 The Determinant Line Bundle.- 10 The Family Index Theorem.- 10.1 Riemannian Fibre Bundles.- 10.2 Clifford Modules on Fibre Bundles.- 10.3 The Bismut Superconnection.- 10.4 The Family Index Density.- 10.5 The Transgression Formula.- 10.6 The Curvature of the Determinant Line Bundle.- 10.7 The Kirillov Formula and Bismut's Index Theorem.- References.- List of Notation.
2,112 citations
Book•
31 May 1987
TL;DR: In this article, the authors present an algebra of continuous linear operators on Hilbert spaces, which is a generalization of the notion of a continuous linear operator on the space L 2 (Rm, Cm).
Abstract: 1. Preliminaries.- 1. Metric Spaces. Normed Spaces.- 2. Algebras and ?-Algebras of Sets.- 3. Countably Additive Functions and Measures.- 4. Measurable Functions.- 5. Integration.- 6. Function Spaces.- 2. Hilbert Space Geometry. Continuous Linear Operators.- 1. Hilbert Space. The Space L2.- 2. Orthonormal Systems.- 3. Projection Theorem. Orthogonal Expansions and Orthogonal Sums.- 4. Linear Functionals and Sesqui-linear Forms. Weak Convergence.- 5. The Algebra of Continuous Operators on H.- 6. Compact Operators.- 7. Bounded Self-adjoint Operators.- 8. Orthogonal Projections.- 9. Examples of Hilbert Spaces and Orthonormal Systems.- 10. Examples of Continuous Functionals and Operators.- 3. Unbounded Linear Operators.- 1. General Notions. Graph of an Operator.- 2. Closed Operators. Closable Operators.- 3. Adjoint Operator.- 4. Domination of Operators.- 5. Invariant Subspaces.- 6. Reducing Subspaces.- 7. Defect Number, Spectrum, and Resolvent of a Closed Operator.- 8. Skew Decompositions. Skew Reducibility.- 9. Spectral Theory of Compact Operators.- 10. Connection between the Spectral Properties of TS and ST.- 4. Symmetric and Isometric Operators.- 1. Symmetric and Self-adjoint Operators. Deficiency Indices.- 2. Isometric and Unitary Operators.- 3. Cayley Transform.- 4. Extensions of Symmetric Operators. Von Neumann's Formulae.- 5. The Operator T*T. Normal Operators.- 6. Classification of Spectral Points.- 7. Multiplication by the Independent Variable.- 8. Differentiation Operator.- 5. Spectral Measure. Integration.- 1. Basic Notions.- 2. Extension of a Spectral Measure. Product Measures.- 3. Integral with Respect to a Spectral Measure. Bounded Functions.- 4. Integral with Respect to a Spectral Measure. Unbounded Functions.- 5. An Example of Commuting Spectral Measures whose Product is not Countably Additive.- 6 Spectral Resolutions.- 1. Statements of Spectral Theorems. Functions of Operators.- 2. Spectral Theorem for Unitary Operators.- 3. Spectral Theorem for Self-adjoint Operators.- 4. Spectral Resolution of a One-parameter Unitary Group.- 5. Joint Spectral Resolution for a Finite Family of Commuting Self-adjoint Operators.- 6. Spectral Resolutions of Normal Operators.- 7 Functional Model and the Unitary Invariants of Self-adjoint Operators.- 1. Direct Integral of Hilbert Spaces.- 2. Multiplication Operators and Decomposable Operators.- 3. Generating Systems and Spectral Types.- 4. Unitary Invariants of Spectral Measure.- 5. Unitary Invariants of Self-adjoint Operators.- 6. Decomposition of a Spectral Measure into the Absolutely Continuous and the Singular Part.- 8 Some Applications of Spectral Theory.- 1. Polar Decomposition of a Closed Operator.- 2. Differential Equations of Evolution on Hilbert Space.- 3. Fourier Transform.- 4. Multiplications on L2 (Rm, Cm).- 5. Differential Operators with Constant Coefficients.- 6. Examples of Differential Operators.- 9 Perturbation Theory.- 1. Essential Spectrum. Compact Perturbations.- 2. Compact Self-adjoint and Normal Operators.- 3. Finite-dimensional Perturbations and Extensions.- 4. Continuous Perturbations.- 10 Semibounded Operators and Forms.- 1. Closed Positive Definite Forms.- 2. Semibounded Forms.- 3. Friedrichs Method of Extension of a Semibounded Operator to a Self-adjoint Operator.- 4. Fractional Powers of Operators. The Heinz Inequality.- 5. Examples of Quadratic Forms. The Sturm-Liouville Operator on [?1, 1].- 6. Examples of Quadratic Forms. One-dimensional Schrodinger Operator.- 11 Classes of Compact Operators.- 1. Canonical Representation and Singular Numbers of Compact Operators.- 2. Nuclear Operators. Trace of an Operator.- 3. Hilbert-Schmidt Operators.- 4. Sp Classes.- 5. Additional Information on Singular Numbers of Compact Operators.- 6. ?p Classes.- 7. Lidskii's Theorem.- 8. Examples of Compact Operators.- 12 Commutation Relations of Quantum Mechanics.- 1. Statement of the Problem. Auxiliary Material.- 2. Properties of (B)-systems and (C)-systems.- 3. Representations of the Bose Relations. The Case m = 1.- 4. Representations of the Bose Relations. General Case.- 5. Representations of the Canonical Relations.
910 citations
TL;DR: In this paper, it was shown that λk is the kth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, and the upper bound of the number of eigenvalues less than or equal to -α for the operator Δ−V(x) defined on Ωn (n≧3) in terms of √ √ n/2 is also provided.
Abstract: If λk is thekth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, H. Weyl's asymptotic formula asserts that\(\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}} \right)^{2/n} \), hence\(\sum\limits_{i = 1}^k {\lambda _i \sim \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} } \). We prove that for any domain and for all\(\sum\limits_{i = 1}^k {\lambda _i \geqq \frac{{nC_n }}{{n + 2}}k^{\frac{{n + 2}}{n}} V(D)^{ - 2/n} } \). A simple proof for the upper bound of the number of eigenvalues less than or equal to -α for the operator Δ−V(x) defined on ℝn (n≧3) in terms of\(\int\limits_{\mathbb{R}^n } {(V + \alpha )_ - ^{n/2} } \) is also provided.
534 citations