Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator

We introduce the Umbral calculus into Clifford analysis starting from the abstract of the Heisenberg commutation relation $[\frac{d}{dx},x]=\mathbf{i}\mathbf{d}$. The Umbral Clifford analysis provides an effective framework in continuity and discreteness. In this paper we consider functions defined in a star-like domain $\Omega \subset \BR^n$withvaluesintheUmbralCliffordalgebra$C\ell_{0,n}'$ which are Umbral polymonogenic with respect to the (left) Umbral Dirac operator $D^{\mathrm{\prime }}$, i.e. they belong to the kernel of $(D^{\mathrm{\prime }}{)}^k$. We prove that any polymonogenic function $f$ has a decomposition of the form $$f=f_1+x^{\mathrm{\prime }}{f}_2+\mathrm{.}\mathrm{.}\mathrm{.}+(x^{\mathrm{\prime }}{)}^{k-1}{f}_k,$$ where $x^{\mathrm{\prime }}=x_1^{\mathrm{\prime }}{e}_1+\mathrm{.}\mathrm{.}\mathrm{.}+x_n^{\mathrm{\prime }}{e}_n$ and $f_j,j=1,\mathrm{.}\mathrm{.}\mathrm{.},k,$ 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.