Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

Rozenblum, Grigori; Shargorodsky, Eugene

doi:10.48550/arxiv.2011.14877

Cited by 3 publications

(18 citation statements)

References 11 publications

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“…We refer to [1,2,42,71,72,73] for various extensions of the above Weyl's law. Note that for p = n the asymptotics (8.16) implies a stronger form of Connes' trace theorem (5.2) (see [71,90]).…”

Section: Borderline Birman-schwinger Principlementioning

confidence: 99%

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

McDonald¹,

Ponge²

2021

Preprint

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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 Lp-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) Schrödinger operators that are built out of fractional Laplace-Beltrami operators and Lp-potentials. We also conjecture a semi-classical Weyl's law for such operators.

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Section: Borderline Birman-schwinger Principlementioning

confidence: 99%

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

McDonald¹,

Ponge²

2021

Preprint

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“…In the recent papers [23], [25], Birman-Schwinger type operators in a domain Ω ⊆ R N were considered, namely, the ones having the form T ≡ T P ≡ T P,A = A * P A. Here A is a pseudodifferential operator in Ω of order −l = −N/2 and P = V μ is a finite signed Borel measure containing a singular part.…”

Section: Introductionmentioning

confidence: 99%

“…The reasoning in [23], [25] was based upon the traditional variational approach to the study of eigenvalue distribution, in the version originating in papers by M.Sh. Birman and M.Z.…”

Section: Introductionmentioning

confidence: 99%

“…Taking into account the results in [23], [25], by our opinion, it is interesting to investigate spectral properties of operators of the form T P with singular a measure P = V μ in the noncritical case, i.e., for l = N/2. Our results in this paper show that the eigenvalue behavior in the subcritical, l < N/2, and in the supercritical, l > N/2, cases differ essentially from each other, and both from the critical case above.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalues of the Birman-Schwinger operator for singular measures: The noncritical case

Rozenblum

Tashchiyan

2022

Journal of Functional Analysis

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“…This is stronger than PT -measurability and implies Theorem 1.1. For more recent work, see Rozenblum and Shargorodsky [59].…”

Section: Introductionmentioning

confidence: 99%

Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

McDonald,

Sukochev,

Zanin

2021

Preprint

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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 (A, H, D) and self-adjoint V ∈ A, there holds

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Eigenvalue estimates and asymptotics for weighted pseudodifferential operators with singular measures in the critical case

Cited by 3 publications

References 11 publications

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

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

Eigenvalues of the Birman-Schwinger operator for singular measures: The noncritical case

Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

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