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 .

Heat kernels and Green functions of sub-Laplacians on Heisenberg groups with multi-dimensional center

We compute the sub-Laplacian on the Heisenberg group with multi-dimensional center. By taking the inverse Fourier transform with respect to the center, we get the parametrized twisted Laplacians. Then by means of the special Hermite functions, we find the eigenfunctions and the eigenvalues of the twisted Laplacians. The explicit formulas for the heat kernels and Green functions of the twisted Laplacians can then be obtained. Then we give an explicit formula for the heat kernal and Green function of the sub-Laplacian on the Heisenberg group with multi-dimensional center.