Abstract Harmonic Analysis

Hewitt, Edwin; Ross, Kenneth A.

doi:10.1007/978-3-662-40409-6

Cited by 602 publications

(640 citation statements)

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“…By contrast, in the polymer Hilbert space the action ofÛ λ is not weakly continuous in λ, and a basic momentum operator does not exist. The states in the polymer Hilbert space can be described as points in a certain compact space, the (Harald) Bohr compactification of the real line, and the operators introduced above can be described in terms of a representation of the Weyl algebra associated with the classical position and momentum variables [11,12,13]. There exists also a mirror-image quantization in which a a momentum operator and a family of translation operators in the momenta exist, but there is no basic position operator [14].…”

Section: Polymer Quantization On Rmentioning

confidence: 99%

Quantum gravity and the Coulomb potential

2007

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We apply a singularity resolution technique utilized in loop quantum gravity to the polymer representation of quantum mechanics on R with the singular −1/|x| potential. On an equispaced lattice, the resulting eigenvalue problem is identical to a finite difference approximation of the Schrödinger equation. We find numerically that the antisymmetric sector has an energy spectrum that converges to the usual Coulomb spectrum as the lattice spacing is reduced. For the symmetric sector, in contrast, the effect of the lattice spacing is similar to that of a continuum self-adjointness boundary condition at x = 0, and its effect on the ground state is significant even if the spacing is much below the Bohr radius. Boundary conditions at the singularity thus have a significant effect on the polymer quantization spectrum even after the singularity has been regularized.

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Section: Polymer Quantization On Rmentioning

confidence: 99%

Quantum gravity and the Coulomb potential

2007

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“…We shall use various well-known facts from abstract harmonic analysis, the structure theory of locally compact abelian groups [9], and the theory of infinite abelian groups [7,8].…”

Section: Introductionmentioning

confidence: 99%

Heyde’s Characterization Theorem for Discrete Abelian Groups

Myronyuk

2010

J. Aust. Math. Soc.

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Let X be a countable discrete abelian group with automorphism group Aut(X ). Let ξ 1 and ξ 2 be independent X -valued random variables with distributions µ 1 and µ 2 , respectively. Suppose that α 1 , α 2 , β 1 , β 2 ∈ Aut(X ) andAssuming that the conditional distribution of the linear form L 2 given L 1 is symmetric, where L 2 = β 1 ξ 1 + β 2 ξ 2 and L 1 = α 1 ξ 1 + α 2 ξ 2 , we describe all possibilities for the µ j . This is a group-theoretic analogue of Heyde's characterization of Gaussian distributions on the real line.2000 Mathematics subject classification: primary 60B15; secondary 62E10.

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“…The linear maps P H and T H have significant roles in abstract harmonic analysis over homogeneous spaces, see [6,13,22]. Corollary 3.4 asserts connections of partial integration over H with P H and T H .…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Abstract Relative Fourier Transforms Over Canonical Homogeneous Spaces of Semi-Direct Product Groups With Abelian Normal Factor

Farashahi¹

2017

Journal of the Korean Mathematical Society

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Abstract. This paper presents a systematic study for theoretical aspects of a unified approach to the abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let H be a locally compact group, K be a locally compact Abelian (LCA) group, and θ : H → Aut(K) be a continuous homomorphism. Let G θ = H ⋉ θ K be the semi-direct product of H and K with respect to θ and G θ /H be the canonical homogeneous space (left coset space) of G θ . We introduce the notions of relative dual homogeneous space and also abstract relative Fourier transform over G θ /H. Then we study theoretical properties of this approach.

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Abstract Harmonic Analysis

Cited by 602 publications

References 0 publications

Quantum gravity and the Coulomb potential

Quantum gravity and the Coulomb potential

Heyde’s Characterization Theorem for Discrete Abelian Groups

Abstract Relative Fourier Transforms Over Canonical Homogeneous Spaces of Semi-Direct Product Groups With Abelian Normal Factor

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