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Fock Space, the Heisenberg Group, Heat Flow, and Toeplitz Operators
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
The Heisenberg group, discussed above, is a nilpotent Lie group which plays a key role in quantum mechanics and in the representation theory of nilpotent Lie groups. It is standard that Lebesgue measure on Cn × R is biinvariant Haar measure on Hn. The representation theory for Hn is well-known. The infinite-dimensional strongly-continuous irreducible unitary representations are traditionally parametrized by non-zero values of “Planck’s constant” and act on L2(Rn, dv). In particular, again following [21], the Schrödinger representation is given, for a in Cn and for p, q, x in Rn, with r in R, f(x) in L2(Rn, dv) and a = p + iq, by [ρtS(p+iq,r)f](x)=eirα−2iαqx+iαpqf(x−p).
The heat kernel of sub-Laplace operator on nilpotent Lie groups of step two
Published in Applicable Analysis, 2021
Der-Chen Chang, Qianqian Kang, Wei Wang
The simplest example of nilpotent Lie group of step two is the Heisenberg group . For isotropic Heisenberg group, the heat kernel was independently studied by Gaveau [8] via probability methods and by Hulanicki [9] via the group Fourier transform and Laguerre functions. Later, Beals and Greiner [10] solved the general case via pseudo-differential operators calculus. In [11], Berenstein, Chang and Tie obtained the heat kernel of sub-Laplace operator via the Laguerre calculus. In their celebrated papers, Beals, Gaveau and Greiner [12,13] constructed the heat kernel and fundamental solution for sub-Laplace operator from geometric point of view and opened a new direction of research in geometric analysis for subelliptic differential operators. Later, Calin, Chang and Markina [14] generalized methods of Beals, Gaveau and Greiner to construct the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two.
A review on some classes of algebraic systems
Published in International Journal of Control, 2020
Víctor Ayala, Heriberto Román-Flores
Let us fix some ideas. In our context, the Lie group G could be the real vector space , a sphere when n=1,3,7, a torus , the Heisenberg group. We also consider any connected matrix group, such as the invertible real matrices of order n, or their subgroups and the matrix groups of determinant 1 and orthogonal, respectively.
Optical modeling for electromagnetic Heisenberg ferromagnetic microscale in Heisenberg group
Published in Waves in Random and Complex Media, 2022
Talat Körpinar, Zeliha Körpinar
Heisenberg magnetical ferromagneticmicroscale in Heisenberg group is By easy calculations, we have where By definition of ferromagnetic magnetic optimistic density in Heisenberg group, we get Heisenberg magnetical ferromagneticmicroscale in Heisenberg group is obtained byElectricalmicroscale in Heisenberg group is given by electroosmotic velocity From definition of electric optimistic density in Heisenberg group, we get Heisenberg electricalmicroscale in Heisenberg group is constructed by By definition of ferromagnetic electric optimistic density in Heisenberg group, we present Heisenberg electrical ferromagnetic microscale foris The optical normalized intensity as magnetic ferromagnetic quasi microscale of evolving time t is presented at Figure 2 for diverse stator field.