Eigenvalue estimates for the Dirac operator with generalized APS boundary condition

Chen, Daguang

doi:10.1016/j.geomphys.2006.03.009

Cited by 6 publications

(5 citation statements)

References 16 publications

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“…All lower bounds for the Dirac operator involve the scalar curvature of M as well as the mean curvature of ∂M , as can be expected because of the central role of the Schrödinger-Lichnerowicz formula relating the squared Dirac operator with the associated rough Laplacian. Moreover, our lower bounds enhance and rely on former ones due to D. Chen [10,Thm. 3.1] for the so-called gAPS boundary condition (generalizing [23,Thm.…”

Section: Introductionmentioning

confidence: 67%

“…In [22], the authors provide a Friedrich-type lower bound involving scalar curvature [15] for the first eigenvalue of the Dirac operator and for each of the above boundary conditions (see also [10], [23]). They also discuss the equality case of those estimates which turns out not to be always achieved depending on the imposed boundary condition; moreover, for the cases where the equality is realized, the boundary has to be minimal.…”

Section: Eigenvalue Estimates For the Dirac Operatormentioning

confidence: 99%

“…3] as well as the MIT bag [22,Thm. 4] boundary conditions, and that of D. Chen again [10,Thm. 3.3] for the so-called mgAPS boundary condition (generalizing [22,Thm.…”

Section: Introductionmentioning

confidence: 97%

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New eigenvalue estimates involving Bessel functions

Chami¹,

Ginoux²,

Habib³

2021

Publicacions Matemàtiques

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Given a compact Riemannian manifold (M n , g) with boundary ∂M , we give an estimate for the quotient ∂M f dµ g M f dµ g, where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a direct proof is given of the Faber-Krahn inequalities for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

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Section: Introductionmentioning

confidence: 67%

Section: Eigenvalue Estimates For the Dirac Operatormentioning

confidence: 99%

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New eigenvalue estimates involving Bessel functions

Chami¹,

Ginoux²,

Habib³

2021

Publicacions Matemàtiques

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“…The APS(Atiyah-Patodi-Singer) boundary condition plays an important role in the index theory for the Dirac operator, the modified APS condition was introduced in [10]. Chen generalized the APS boundary condition in [3].…”

Section: Boundary Conditionsmentioning

confidence: 99%

Eigenvalue estimates via Hömander's $L^2$-method

Ji,

Lin

2019

Preprint

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“…Then the operator L should be replaced by L φ in Theorem 3 and Theorem 6 in [19] (It would be interesting to discuss other types of boundary conditions [14,13,8] and to generalize these results to the case m > 1).…”

Section: Boundary Value Problemsmentioning

confidence: 99%