Quantum topology impact factor

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In this article, we analyze the spectrum of discrete magnetic Laplacians [DML] on an infinite covering graphG˜G=G˜/Γwith [Abelian] lattice groupΓand periodic magnetic potentialβ˜. We give sufficient conditions for [...] Read more.

In this article, we analyze the spectrum of discrete magnetic Laplacians [DML] on an infinite covering graphG˜G=G˜/Γwith [Abelian] lattice groupΓand periodic magnetic potentialβ˜. We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend onβ˜. The magnetic potential can be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers [polyacetylene] and nanoribbons in the presence of a constant magnetic field. Full article

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An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the LaplaceBeltrami operator over a [...] Read more.

An analysis of the invariance properties of self-adjoint extensions of symmetric operators under the action of a group of symmetries is presented. For a given group G, criteria for the existence of G-invariant self-adjoint extensions of the LaplaceBeltrami operator over a Riemannian manifold are illustrated and critically revisited. These criteria are employed for characterising self-adjoint extensions of the LaplaceBeltrami operator on an infinite set of intervals,Ω, constituting a quantum circuit, which are invariant under a given action of the groupZ. A study of the different unitary representations of the groupZon the space of square integrable functions onΩis performed and the correspondingZ-invariant self-adjoint extensions of the LaplaceBeltrami operator are introduced. The study and characterisation of the invariance properties allows for the determination of the spectrum and generalised eigenfunctions in particular examples. Full article

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An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a [...] Read more.

An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for2[G]. An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples. Full article

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In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. [...] Read more.

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschkes theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnsides theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwingers description of quantum mechanical systems. Full article