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Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Hafedh Herichi, Michel L. Lapidus (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

We survey some of the universality properties of the Riemann zeta function ζ ( s ) and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator 𝔞 (mapping...

Two-parameter non-commutative Central Limit Theorem

Natasha Blitvić (2014)

Annales de l'I.H.P. Probabilités et statistiques

In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and q -deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a * -algebra equipped with a state) drawn from an ensemble of pair-wise commuting...

Wehrl entropy of the state in a two-atom Tavis-Cummings model

Debraj Nath, P. K. Das (2011)

Banach Center Publications

In this paper we present an entropic description of quantum state obtained by interaction of one mode of quantized electromagnetic field with two two-level atoms inside a cavity, known as Tavis-Cumming model. Wehrl entropy has been calculated analytically and investigated as a function of the average value of the photon number operator. Husimi's Q function has been calculated and compared with the behaviour of the field entropy.

Weyl quantization for the semidirect product of a compact Lie group and a vector space

Benjamin Cahen (2009)

Commentationes Mathematicae Universitatis Carolinae

Let G be the semidirect product V ⋊ K where K is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space V . Let 𝒪 be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation π of G . We consider the case when the corresponding little group H is the centralizer of a torus of K . By dequantizing a suitable realization of π on a Hilbert space of functions on ℂ n where n = dim ( K / H ) , we construct a symplectomorphism between...

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