Hermite functions and special Hermite functions depending on Heisenberg group (ℍⁿ ): دوال هرميت ودوال هرميت الخاصة اعتماداً على زمرة هايزنبرغ (ℍⁿ)

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we talk about the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operator and momentum operators (ors). relationship between the representation theory of the Heisenberg group and the position and momentum, that shows how we will make the connection between the Heisenberg group and physics. Then we introduce and study some properties of the Hermite and special Hermite functions. These functions are eigenfunctions of the Hermite and special Hermite operators, respectively. The Hermite operator is often called the harmonic oscillator and the special Hermite operator is sometimes called the twisted Laplacian. As we will later see, the two operators are directly related to the sub-laplacian on the Heisenberg group. The theory of Hermite and special Hermite expansions is intimately connected to the harmonic analysis on the Heisenberg group. They play an important role in our understanding of several problems on ℍⁿ .

A Hardy–Sobolev inequality for the twisted Laplacian

We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn. The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.

On Concrete Spectral Properties of a Twisted Laplacian Associated with a Central Extension of the Real Heisenberg Group

We consider the special magnetic Laplacian given byΔν,μ=4∑j=1n∂2/∂zj∂zj¯+2iνE+E¯+n+2μE-E¯-ν2+μ2z2. We show thatΔν,μis connected to the sub-Laplacian of a group of Heisenberg type given byC ×ω Cnrealized as a central extension of the real Heisenberg groupH2n+1. We also discuss invariance properties ofΔν,μand give some of their explicit spectral properties.