scholarly journals Eigenvalue bounds for the Dirac operator

1986 ◽  
Vol 125 (1) ◽  
pp. 117-126 ◽  
Author(s):  
John Lott
Keyword(s):  
Dirac Operator ◽  
Eigenvalue Bounds

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

scholarly journals DIRAC OPERATORS ON LIE ALGEBROIDS

2021 ◽  
pp. 983
Author(s):  
Arezo Tarviji ◽  
Morteza Mirmohammad Rezaei
Keyword(s):  
Dirac Operator ◽  
Spin Structure ◽  
Dirac Operators ◽  
Lie Algebroids ◽  
Lie Algebroid ◽  
Eigenvalue Bounds ◽  
Base Manifold ◽  
Spinor Bundle ◽  
Complex Spin ◽  
The One

We compare the Dirac operator on transitive Riemannian Lie algebroid equipped by spin or complex spin structure with the one defined on to its base manifold‎. Consequently we derive upper eigenvalue bounds of Dirac operator on base manifold of spin Lie algebroid twisted with the spinor bundle of kernel bundle‎.

scholarly journals Extended supersymmetries and the Dirac operator

2005 ◽  
Vol 315 (2) ◽  
pp. 467-487 ◽  
Author(s):  
A. Kirchberg ◽  
J.D. Länge ◽  
A. Wipf
Keyword(s):  
Dirac Operator

scholarly journals Computable eigenvalue bounds for rank-k perturbations

2010 ◽  
Vol 432 (12) ◽  
pp. 3100-3116 ◽  
Author(s):  
Jan H. Brandts ◽  
Ricardo Reis da Silva
Keyword(s):  
Eigenvalue Bounds

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

scholarly journals The spectrum of the Dirac operator on SU2/Q8

2008 ◽  
Vol 125 (3) ◽  
pp. 383-409 ◽  
Author(s):  
Nicolas Ginoux
Keyword(s):  
Dirac Operator
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