scholarly journals A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Asao Arai
Keyword(s):  
Dirac Operator ◽  
Hilbert Spaces ◽  
Special Class ◽  
Spectral Properties ◽  
Fock Space ◽  
Dirac Operators ◽  
Fredholm Property ◽  
Perturbation Parameter ◽  
Coupling Constant ◽  
Infinite Dimensional

Spectral properties of a special class of infinite dimensional Dirac operatorsQ(α)on the abstract boson-fermion Fock spaceℱ(ℋ,𝒦)associated with the pair(ℋ,𝒦)of complex Hilbert spaces are investigated, whereα∈Cis a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operatorQ(0)is taken to be a free infinite dimensional Dirac operator. A variety of the kernel ofQ(α)is shown. It is proved that there are cases where, for all sufficiently large|α|withα<0,Q(α)has infinitely many nonzero eigenvalues even ifQ(0)has no nonzero eigenvalues. Also Fredholm property ofQ(α)restricted to a subspace ofℱ(ℋ,𝒦)is discussed.

scholarly journals Products of Positive Operators

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Maximiliano Contino ◽  
Michael A. Dritschel ◽  
Alejandra Maestripieri ◽  
Stefania Marcantognini
Keyword(s):  
Positive Operator ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Compact Operators ◽  
Positive Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Special Classes

AbstractOn finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

scholarly journals An Abstract Linking Theorem Applied to Indefinite Problems Via Spectral Properties

2019 ◽  
Vol 19 (3) ◽  
pp. 545-567 ◽  
Author(s):  
Liliane A. Maia ◽  
Mayra Soares
Keyword(s):  
Hilbert Spaces ◽  
Critical Points ◽  
Spectral Properties ◽  
Schrodinger Equations ◽  
Linking Theorem ◽  
Schrödinger Equations ◽  
Infinite Dimensional ◽  
Cerami Condition ◽  
Indefinite Problems ◽  
Cerami Sequences

Abstract An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite-dimensional Hilbert Spaces. The applications are given to Schrödinger equations. Here spectral properties inherited by the potential features are exploited in order to establish a linking structure, and hence hypotheses of monotonicity on the nonlinearities are discarded.

Channel Covariance Conversion and Modelling Using Infinite Dimensional Hilbert Spaces

2021 ◽  
pp. 1-1
Author(s):  
Lorenzo Miretti ◽  
Renato Calvacante ◽  
Slawomir Stanczak
Keyword(s):  
Hilbert Spaces ◽  
Infinite Dimensional

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 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 ON EIGENVALUE ESTIMATES FOR THE SUBMANIFOLD DIRAC OPERATOR

2002 ◽  
Vol 13 (05) ◽  
pp. 533-548 ◽  
Author(s):  
NICOLAS GINOUX ◽  
BERTRAND MOREL
Keyword(s):  
Dirac Operator ◽  
Lower Bounds ◽  
Dirac Operators ◽  
Extrinsic Curvature ◽  
Eigenvalue Estimates ◽  
Killing Spinors ◽  
Spinor Fields ◽  
Twisted Dirac Operators

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

scholarly journals Infinite-dimensional integration on weighted Hilbert spaces

2012 ◽  
Vol 81 (280) ◽  
pp. 2175-2205 ◽  
Author(s):  
Michael Gnewuch
Keyword(s):  
Hilbert Spaces ◽  
Infinite Dimensional

Teleportation schemes in infinite dimensional Hilbert spaces

2005 ◽  
Vol 46 (10) ◽  
pp. 102103 ◽  
Author(s):  
Karl-Heinz Fichtner ◽  
Wolfgang Freudenberg ◽  
Masanori Ohya
Keyword(s):  
Hilbert Spaces ◽  
Infinite Dimensional
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