令Q=X×H是一个四元数Heisenberg群, 其中,X是一个2×2的Pauli矩阵. 文章①给出四元数Heisenberg群Q的薛定谔表示;②通过Weyl变换研究四元数Heisenberg群Q上的奇异卷积算子, 结合奇异卷积算子的性质得到了Radon变换的逆公式;③得到了Radon变换是索伯列夫空间W到L2(Q)的有界酉算子.
Abstract
Let Q=X×H be the quaternion Heisenberg group, where X denotes the set of all 2×2 Pauli matrices. In this article, we first define the Schrödinger representation on the quaternion Heisenberg group Q. Moreover, we deal with the singular convolution operator on Q by Weyl transform. At length, we obtain the inverse formula of Radon transform, and prove that the Radon transform on Q is a bounded unitary operator from a Sobolev space W into L2(Q).
关键词
四元数Heisenberg群 /
薛定谔表示 /
Radon变换 /
奇异卷积算子
{{custom_keyword}} /
Key words
quaternion Heisenberg group /
Schrödinger representation /
Radon transform /
singular convolution operator
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] LIBOFF R L. Introductory quantum mechanics[M]. San Francisco: Addison-Wesley, 2002.
[2] BARKER S R, SALAMON S M. Analysis on a generalized heisenberg group[J]. Journal of the London Mathematical Society, 1983, 28(1): 184-192.
[3] TIE J Z, WONG M W. The heat kernel and green functions of sub-laplacians on the quaternion heisenberg group[J]. Journal of Geometric Analysis, 2009, 19(1): 191-210.
[4] DEITMAR A. A first course in harmonic analysis[M]. 2nd ed. New York: Universitext Springer-Verlag, 2005.
[5] PENG L Z, ZHAO J M. Weyl transforms associated with the Heisenberg group[J]. Bulletin des Sciences Mathématiques, 2008, 132(1): 78-86.
[6] WONG M W. Weyl transforms[M]. New York: Universitext Springer-Verlag, 1998.
[7] GELLER D, STEIN E M. Singular convolution operators on the Heisenberg group[J]. Mathematische Annalen, 1984, 267(1): 1-15.
[8] HE J X, LIU H P. Admissible wavelets and inverse Radon transform associated with the affine homogeneous Siegel domains of type II[J]. Communications in Analysis and Geometry, 2007, 1(1): 1-28.
[9] HE J X, LIU H P. Wavelet transform and Radon transform on the quaternion Heisenberg group[J]. Acta Mathematica Sinica (English Ser), 2014, 30(4): 619-636.
[10]COWLING M, DOOLEY A H, KORÁNYI A. H-type groups and Iwasawa decompositions[J]. Advances in Mathematics, 1991, 87(1):1-41.
[11]李洪全. Heisenberg型群上的强奇异卷积算子[J]. 中国科学:数学, 2014(5):497-508.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
基金
国家自然科学基金资助项目(11471040;11671414)
{{custom_fund}}
PDF(1060 KB)
