Functional calculus for elliptic operators on noncommutative tori, I

Lee, Gihyun; Ponge, Raphaël

doi:10.1007/s11868-020-00337-z

Cited by 7 publications

(7 citation statements)

References 25 publications

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“…Thus, our results confirm the claim of [12]. In any case, we postpone to [27] a more thorough account on complex powers of elliptic ΨDOs on noncommutative tori.…”

Section: Introductionsupporting

confidence: 87%

“…In this section, we use the results of the previous section to show that the complex powers of positive elliptic ΨDOs are ΨDOs. We refer to [27] for a more comprehensive account on complex powers of elliptic ΨDOs on noncommutative tori.…”

Section: Complex Powers Of Positive Elliptic Operatorsmentioning

confidence: 99%

“…In [27] we shall apply the above results to construct complex powers P s of an elliptic ΨDO as holomorphic families of ΨDOs. As a preview of this we show when P P Ψ w pA θ q is an elliptic ΨDO with positive principal symbol and positive spectrum its complex powers P z , z P C, are ΨDOs (Theorem 7.3).…”

Section: Introductionmentioning

confidence: 99%

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Functional Calculus for Elliptic Operators on Noncommutative Tori, I

Lee¹,

Ponge²

2019

Preprint

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In this paper, we introduce a parametric pseudodifferential calculus on noncommutative n-tori which is a natural nest for resolvents of elliptic pseudodifferential operators. Unlike in some previous approaches to parametric pseudodifferential calculi, our parametric pseudodifferential calculus contains resolvents of elliptic pseudodifferential operators that need not be differential operators. As an application we show that complex powers of positive elliptic pseudodifferential operators on noncommutative n-tori are pseudodifferential operators. This confirms a claim of Fathi-Ghorbanpour-Khalkhali.

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“…Thus, our results confirm the claim of [12]. In any case, we postpone to [27] a more thorough account on complex powers of elliptic ΨDOs on noncommutative tori.…”

Section: Introductionsupporting

confidence: 87%

Section: Complex Powers Of Positive Elliptic Operatorsmentioning

confidence: 99%

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Functional Calculus for Elliptic Operators on Noncommutative Tori, I

Lee¹,

Ponge²

2019

Preprint

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“…In particular, it follows from (6.5) that ρ 2 (ξ) is positive and invertible for all ξ ∈ R n \ 0. All this allows us to apply [46,Theorem 7.3]. Thus, for every s ∈ R, the power ∆s g,ν is an operator in Ψ 2s (T n θ ) with principal symbol |ξ| 2s g .…”

Section: Curved Cwikel Estimates and Integration Formulasmentioning

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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“…Given any symbol ρ(ξ) ∈ S m R n ; A θ denote byρ(ξ) the symbol (3.3) associated with ρ k = ρ(k), k ∈ Z n . By [33,Lemma 5.21] this gives rise to a continuous linear map ρ(ξ) →ρ(ξ) − ρ(ξ) from S m R n ; A θ to S R n ; A θ . It then follows that (ρ(z)−ρ(z)) z∈Ω is a holomorphic family in S R n ; A θ .…”

Section: Canonical Tracementioning

confidence: 99%

Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results

Ponge¹

2020

SIGMA

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In this paper we establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators.The latter is the unique trace up to constant multiple on noninteger order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and Lévy-Neira-Paycha.

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Functional calculus for elliptic operators on noncommutative tori, I

Cited by 7 publications

References 25 publications

Functional Calculus for Elliptic Operators on Noncommutative Tori, I

Functional Calculus for Elliptic Operators on Noncommutative Tori, I

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

Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results

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