The Dirac Spectrum

Ginoux, Nicolas

doi:10.1007/978-3-642-01570-0

Cited by 110 publications

(141 citation statements)

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“…By comparing with the Schrödinger-Lichnerowicz formula (8)- (9) we see that the operator (U * 0 U * LUU 0 ) ⊕ (U * LU) is unitarily equivalent to (D Σ ) 2 . As (D Σ ) 2 is now unitarily equivalent to D / 2 ⊕ D / 2 (because n is even), it follows that L is unitarily equivalent to D / 2 .…”

Section: Intrinsic and Extrinsic Dirac Operators On Euclidean Hypersumentioning

confidence: 96%

Dirac Operators on Hypersurfaces as Large Mass Limits

Moroianu

Ourmières-Bonafos

Pankrashkin

2019

Commun. Math. Phys.

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Section: Intrinsic and Extrinsic Dirac Operators On Euclidean Hypersumentioning

confidence: 96%

Dirac Operators on Hypersurfaces as Large Mass Limits

Moroianu

Ourmières-Bonafos

Pankrashkin

2019

Commun. Math. Phys.

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“…The spectrum of the intrinsic Dirac operator has also been studied [Gin09]. The Dirac operator is the most fundamental elliptic operator on manifolds [Esp98, section 3.6], indicating that Dirac spectra are supposed to contain more information than the Laplacian spectra (for evidence see Fig.…”

Section: Related Workmentioning

confidence: 99%

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Diamanti

Tang

et al. 2018

Computer Graphics Forum

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Spectral mesh analysis and processing methods, namely ones that utilize eigenvalues and eigenfunctions of linear operators on meshes, have been applied to numerous geometric processing applications. The operator used predominantly in these methods is the Laplace‐Beltrami operator, which has the often‐cited property that it is intrinsic, namely invariant to isometric deformation of the underlying geometry, including rigid transformations. Depending on the application, this can be either an advantage or a drawback. Recent work has proposed the alternative of using the Dirac operator on surfaces for spectral processing. The available versions of the Dirac operator either only focus on the extrinsic version, or introduce a range of mixed operators on a spectrum between fully extrinsic Dirac operator and intrinsic Laplace operator. In this work, we introduce a unified discretization scheme that describes both an extrinsic and intrinsic Dirac operator on meshes, based on their continuous counterparts on smooth manifolds. In this discretization, both operators are very closely related, and preserve their key properties from the smooth case. We showcase various applications of our operators, with improved numerics over prior work.

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“…(1.3)). For more information about the first Dirac eigenvalue and generalizations of the Killing spinor equation, we refer the readers to references [4,5,8,9,10,11,12]. We now turn to the first Dirac eigenvalue problem on Sasakian spin manifolds.…”

Section: Introductionmentioning

confidence: 99%

Sasakian Twistor Spinors and the First Dirac Eigenvalue

Kim¹

2016

Journal of the Korean Mathematical Society

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Abstract. On a closed eta-Einstein Sasakian spin manifold of dimension 2m + 1 ≥ 5, m ≡ 0 mod 2, we prove a new eigenvalue estimate for the Dirac operator. In dimension 5, the estimate is valid without the eta-Einstein condition. Moreover, we show that the limiting case of the estimate is attained if and only if there exists such a pair (ϕ m 2 −1 , ϕ m 2 ) of spinor fields (called Sasakian duo, see Definition 2.1) that solves a special system of two differential equations.

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The Dirac Spectrum

Cited by 110 publications

References 0 publications

Dirac Operators on Hypersurfaces as Large Mass Limits

Dirac Operators on Hypersurfaces as Large Mass Limits

A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing

Sasakian Twistor Spinors and the First Dirac Eigenvalue

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