A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

In this note, we use Schrödinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol, which is associated with the rescaled harmonic oscillator. Then we can give an explicit formula for the heat kerne...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2022-09, Vol.38 (9), p.1597-1611
Main Author: Yang, Zhi Peng
Format: Article
Language:English
Subjects:
Fourier transforms
Harmonic oscillators
Kernels
Lie groups
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
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Summary:In this note, we use Schrödinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol, which is associated with the rescaled harmonic oscillator. Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin. Our results are useful for the general two step nilpotent Lie groups, including the Heisenberg group and H-type group.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-022-2100-8