Uniqueness theorems for the impulsive Dirac operator with discontinuity

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.

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A Discrete Analogue of the Dirac Operator and the Discrete Modified Novikov–Veselov Hierarchy

International Mathematics Research Notices ◽

10.1093/imrn/rnq014 ◽

2010 ◽

Vol 2010 (18) ◽

pp. 3463-3488 ◽

Cited By ~ 2

Author(s):

Dmitry Zakharov

Keyword(s):

Dirac Operator ◽

Discrete Analogue

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Higher spectral flow and an entire bivariant JLO cocycle

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is012008031jkt193 ◽

2012 ◽

Vol 11 (1) ◽

pp. 183-232 ◽

Cited By ~ 5

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.

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