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- W196408376 abstract "The Dirac operator on the Euclidean space ℝ n , n ≥ 2, is a first-order differential operator ( mathfrak{D}_{euc,n} ) with coefficients in the real Clifford algebra ( mathfrak{A}_{euc,n} ) associated with ℝ n that has the defining property ( mathfrak{D}_{euc,n}^2 = - Delta _{euc,n} ), where Δeuc,n stands for the standard Laplace operator on ℝ n . As generalizations of this class of operators, we investigate pairs (( mathfrak{D},mathfrak{D}^dag )) of first-order homogeneous differential operators on ℝ n with coefficients in a real Banach algebra ( mathfrak{A} ), such that ( mathfrak{D}mathfrak{D}^dag = mu _L Delta _{euc,n} ) and ( mathfrak{D}^dag mathfrak{D} = mu _R Delta _{euc,n} ), or ( mathfrak{D}^dag mathfrak{D} + mathfrak{D}^dag mathfrak{D} = mu Delta _{euc,n} ), where μL, μR, or μ are some elements of ( mathfrak{A} ). Every pair (( mathfrak{D},mathfrak{D}^dag )) that has the former property is called a Dirac pair of differential operators, and every pair (( mathfrak{D},mathfrak{D}^dag )) with the latter property is called a semi-Dirac pair. Our goal is to prove that for any Dirac, or semi-Dirac pair, (( mathfrak{D},mathfrak{D}^dag )), there are two interrelated Cauchy-Pompeiu type, and, respectively, two Bochner-Martinelli-Koppelman type integral representation formulas, one for ( mathfrak{D} ) and another for ( mathfrak{D}^dag ). In addition, we show that the existence of such integral representation formulas characterizes the two classes of pairs of differential operators." @default.
- W196408376 created "2016-06-24" @default.
- W196408376 creator A5071682707 @default.
- W196408376 date "2010-01-01" @default.
- W196408376 modified "2023-09-25" @default.
- W196408376 title "Deconstructing Dirac operators. III: Dirac and semi-Dirac pairs" @default.
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- W196408376 doi "https://doi.org/10.1007/978-3-0346-0161-0_14" @default.
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