Mathematics – Classical Analysis and ODEs
Scientific paper
2009-01-29
Mathematics
Classical Analysis and ODEs
22 pages
Scientific paper
We introduce the Umbral calculus into Clifford analysis starting from the abstract of the Heisenberg commutation relation $[\frac{d}{dx}, x] = {\bf id}$. The Umbral Clifford analysis provides an effective framework in continuity and discreteness. In this paper we consider functions defined in a star-like domain $\Omega \subset \BR^n$ with values in the Umbral Clifford algebra $C\ell_{0,n}'$ which are Umbral polymonogenic with respect to the (left) Umbral Dirac operator $D'$, i.e. they belong to the kernel of $(D')^k$. We prove that any polymonogenic function $f$ has a decomposition of the form $$f=f_1+ x'f_2 + ... + (x')^{k-1}f_k,$$ where $x'=x'_1e_1 + ... + x'_ne_n$ and $f_j, j=1,..., k,$ are Umbral monogenic functions. As examples, this result recoveries the continuous version of the classical Almansi theorem for derivatives and establishes the discrete version of Almansi theorem for difference operator. The approach also provides a similar result in quantum field about Almansi decomposition related to Hamilton operators. Some concrete examples will presented for the discrete analog version of Almansi Decomposition and for the quantum harmonic oscillator.
Faustino Nelson
Ren Guangbin
No associations
LandOfFree
Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-13693