On the essential spectrum of magnetic pseudodifferential operators

Măntoiu, Marius; Purice, Radu; Richard, Serge

doi:10.1016/j.crma.2006.11.001

Cited by 18 publications

(36 citation statements)

References 15 publications

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“…To give a better idea of the meaning of holes death, let us consider some example. The smallest cycle in H 1 is an empty triangle, one of such cycles is given by the three simplices {(Schur's lemma, Stone-Von Neumann Theorem), (Schur's lemma, Spectral Theorem), (Spectral Theorem, Stone-Von Neumann Theorem)} which is killed by a 2-simplex when the three concepts appear together in the paper [20]. Another interesting example is a 5-step long cycle in [13].…”

Section: Resultsmentioning

confidence: 99%

Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge

et al. 2018

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In the last years complex networks tools contributed to provide insights on the structure of research, through the study of collaboration, citation and co-occurrence networks. The network approach focuses on pairwise relationships, often compressing multidimensional data structures and inevitably losing information. In this paper we propose for the first time a simplicial complex approach to word co-occurrences, providing a natural framework for the study of higher-order relations in the space of scientific knowledge. Using topological methods we explore the conceptual landscape of mathematical research, focusing on homological holes, regions with low connectivity in the simplicial structure. We find that homological holes are ubiquitous, which suggests that they capture some essential feature of research practice in mathematics. k-dimensional holes die when every concept in the hole appears in an article together with other k+1 concepts in the hole, hence their death may be a sign of the creation of new knowledge, as we show with some examples. We find a positive relation between the size of a hole and the time it takes to be closed: larger holes may represent potential for important advances in the field because they separate conceptually distant areas. We provide further description of the conceptual space by looking for the simplicial analogs of stars and explore the likelihood of edges in a star to be also part of a homological cycle. We also show that authors’ conceptual entropy is positively related with their contribution to homological holes, suggesting that polymaths tend to be on the frontier of research.

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Section: Resultsmentioning

confidence: 99%

Co-occurrence simplicial complexes in mathematics: identifying the holes of knowledge

et al. 2018

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“…To include a variable magnetic field, one must introduce a "twisted" form of the Weyl calculus [13,14,15], obtaining a quantization of observables which is invariant under gauge transformations. This realization is based on twisting both the pseudo-differential calculus and the crossed products algebras by a 2-cocycle defined on the group R n with values in the unitary elements of the algebra A .…”

Section: Introductionmentioning

confidence: 99%

“…Hopefully, in some subsequent publication, we will restrict to smaller classes of groups allowing a deeper analytical investigation, involving more realistic spaces of functions. For G = R n and for magnetic-type cocycles this has been undertaken in [11,12] and spectral results for magnetic Hamiltonians are contained in [14].Let us describe the content. We start exposing briefly the C * -algebraic formalism of twisted crossed products and its representation theory.…”

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confidence: 99%

Twisted pseudo-differential operator on type I locally compact groups

Bustos¹,

Măntoiu²

2016

Illinois J. Math.

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Let G be a locally compact group satisfying some technical requirements and G its unitary dual. Using the theory of twisted crossed product C * -algebras, we develop a twisted global quantization for symbols defined on G × G and taking operator values. The emphasis is on the representation-theoretic aspect. For nilpotent Lie groups, the connection is made with a scalar quantization of the cotangent bundle T * (G) and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields. * 2010 Mathematics Subject Classification: Primary 81S30, 47G30, Secondary 22D10, 22D25.The present article generalizes the twisted pseudo-differential operators from the Abelian case [14] to non-commutative phase space. The main obstacle is the fact that, when G is not Abelian, the unitary dual G is no longer a group. It still has a a natural (but complicated) measure theory with respect to which a non-commutative version of Plancherel theorem holds. The cohomological twisting will be shown to be possible in this context.On the other hand, we also extend the highly non-commutative formalism introduced in [16] to the presence of a group 2-cocycle. The constructions in [16] have been predated by the intensive study of pseudo-differential operators on particular types of non-commutative Lie groups, as compact or nilpotent. We cite the articles [5,7,20,21] and the books [6,19]; they contain many other relevant references.The present setting of second countable unimodular type I groups is quite general (besides compact and nilpotent, it contains the Abelian, exponentially solvable or semisimple groups, motion groups and certain discrete groups). In particular, some of them do not possess a Lie structure. Hopefully, in some subsequent publication, we will restrict to smaller classes of groups allowing a deeper analytical investigation, involving more realistic spaces of functions. For G = R n and for magnetic-type cocycles this has been undertaken in [11,12] and spectral results for magnetic Hamiltonians are contained in [14].Let us describe the content. We start exposing briefly the C * -algebraic formalism of twisted crossed products and its representation theory. The standard version can be found in [2,17]. Actually we present a modified version, containing a parameter τ connected to ordering issues. Then we recall some basics things about the cohomology of groups G with coefficients in an Abelian Polish group U . The most interesting case is U = C(G, T), the group of continuous functions on the locally compact group G with values in the torus; it is related with the unitary multipliers of the C * -algebras we deal with. We describe 2-cocycles and their pseudo-trivializations which are related, in the case of R n , with vector potentials corresponding to a given magnetic field.We go on with concrete realizations of the crossed products. We consider C * -algebras composed of functions which are bounded and uniformly continuous over G and invariant under translations. Compatible data are defined as couples (A, γ)...

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“…Starting from a twisted C * -dynamical system, one can construct twisted crossed product C * -algebras [19,20,15] (see also references therein). Let L 1 (X ; A) be the complex vector space of A-valued Bochner integrable functions on X and L 1 -norm…”

Section: Magnetic Twisted Crossed Productsmentioning

confidence: 99%