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Some Differential Operators Related to the Non–Isotropic Heisenberg Sub–Laplacian
Author(s) -
Chang Der–Chen,
Tie Jingzhi
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200101)221:1<19::aid-mana19>3.0.co;2-h
Subject(s) - mathematics , differential operator , laplace operator , isotropy , operator (biology) , heisenberg group , laguerre polynomials , kernel (algebra) , lie algebra , pure mathematics , heat kernel , basis (linear algebra) , mathematical physics , algebra over a field , mathematical analysis , quantum mechanics , physics , geometry , biochemistry , chemistry , repressor , transcription factor , gene
Let ℒ α =$ -{1 \over 2} \sum _{j=1}^n ({\bf Z}_j {\bf {\bar Z}}_j + {\bf {\bar Z}}_j {\bf Z}_j) $ + iα T be the sub–Laplacian on the non–isotropic Heisenberg group H n where Z j ${\bf {\overline Z}}_j$ for j = 1, 2, …, n and T are a basis of the Lie algebra n . We apply the Laguerre calculus to obtain the fundamental solution of the heat kernel exp{— s ℒ α }, the Schrödinger operator exp{— is ℒ α } and the operator $\Delta _{\lambda, \alpha} = -{1 \over 2} \sum _{j=1}^n ({\bf Z}_j {\bf {\bar Z}}_j + {\bf {\bar Z}}_j {\bf {\bar Z}}_j)$ + iα T . We also discuss some basic properties of the wave operator.