Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator

TL;DR

This paper extends Almansi theorems within Umbral Clifford analysis, connecting continuous, discrete, and quantum frameworks, and provides new decompositions for polymonogenic functions related to difference operators and quantum harmonic oscillators.

Contribution

It introduces Umbral calculus into Clifford analysis, establishing Almansi-type decompositions for polymonogenic functions in continuous, discrete, and quantum contexts.

Findings

Recovered classical Almansi theorem in continuous setting

Established discrete Almansi theorem for difference operators

Presented examples related to quantum harmonic oscillator

Abstract

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.