# Operator Ordering and Solution of Pseudo-Evolutionary Equations

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## Abstract

**:**

## 1. Introduction

- Finding an eigenfunction of the ${\widehat{D}}_{t}$ operator, such that$${\widehat{D}}_{t}E(\lambda t)=\lambda E(\lambda t)\phantom{\rule{0.166667em}{0ex}}.$$
- Constructing a pseudo evolution operator (PEO) as$$\widehat{U}(t)=E(\alpha \phantom{\rule{0.277778em}{0ex}}t\phantom{\rule{0.277778em}{0ex}}{\widehat{O}}_{x})\phantom{\rule{0.166667em}{0ex}},$$$$F(x,t)=E(\alpha \phantom{\rule{0.277778em}{0ex}}t\phantom{\rule{0.277778em}{0ex}}{\widehat{O}}_{x})f(x)\phantom{\rule{0.166667em}{0ex}}.$$
- Establishing rules that permit the explicit evaluation of the action of the PEO $\widehat{U}(t)$ on the initial function $f(x)$ in the formal solution Equation (4).

## 2. Laguerre Derivative, Laguerre Exponential and Operator-Ordering

- We specialize the operators in Equation (1) to$${\widehat{D}}_{t}={}_{l}{\partial}_{t}={\partial}_{t}t\phantom{\rule{0.277778em}{0ex}}{\partial}_{t}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}{\widehat{O}}_{x}={\partial}_{x}\phantom{\rule{0.166667em}{0ex}},$$
- The eigenfunction of the Laguerre derivative operator is the Bessel-like function ${}_{l}e(x)$ [14],$${}_{l}e(x)=\sum _{r\ge 0}\frac{{x}^{r}}{{(r!)}^{2}}\phantom{\rule{0.166667em}{0ex}},$$
- In view of explicit computations, it will prove advantageous to express ${}_{l}e(x)$ via an umbral image [15,16] (where we refer to Appendix A for the explicit definition of the full formalism)$${}_{l}e(x)=\widehat{\mathbb{I}}\left(\right)open="("\; close=")">v{e}^{vx}$$Here, v is a formal variable, and $\widehat{\mathbb{I}}$ a formal integration operator, which acts according to$$\widehat{\mathbb{I}}\left(\right)open="("\; close=")">{v}^{\alpha}$$

- We define the auxiliary operators$$\widehat{X}:=-\alpha vt\widehat{x}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\widehat{Y}:=\beta vt{\partial}_{x}\phantom{\rule{0.166667em}{0ex}}.$$
- Applying the Weyl disentanglement rule (taking advantage of the fact that $[[\widehat{X},\widehat{Y}],\widehat{X}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}[[\widehat{X},\widehat{Y}],\widehat{Y}]=0$), we find that$${e}^{\widehat{X}+\widehat{Y}}={e}^{-\frac{1}{2}[\widehat{X},\widehat{Y}]}{e}^{\widehat{X}}{e}^{\widehat{Y}}\phantom{\rule{0.166667em}{0ex}}.$$
- We then eventually arrive at the closed-form expression$$F(x,t)=\widehat{\mathbb{I}}\left(\right)open="("\; close=")">v{e}^{-\frac{{(vt)}^{2}}{2}\alpha \beta}{e}^{-vt\alpha \widehat{x}}{e}^{vt\beta {\partial}_{x}}f(x)\phantom{\rule{0.166667em}{0ex}}.$$

## 3. Pseudo-Evolutive Problems and Matrix Calculus

## 4. Time-Ordering and Concluding Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Umbral Image Type Technique

**Definition**

**A1**

**.**Let $\mathcal{A}=\left\{\lambda \right\}\uplus \mathcal{U}\uplus \mathcal{V}\uplus \mathcal{X}$ be an alphabet of formal variables, where ⊎ denotes the operation of disjoint union, and where $\left\{\lambda \right\}$, $\mathcal{U}$, $\mathcal{V}$ and $\mathcal{X}$ are four (disjoint) alphabets of auxiliary formal variables. We will typically employ notations such as $\mathcal{X}=\{x,y,{x}_{1},{x}_{2},\cdots \}$, where we make use of the indexed variable notations in case of many variables for convenience. Let furthermore ${\mathcal{A}}_{\u2022}=\mathcal{A}\setminus \left\{\lambda \right\}$.

## Appendix B. An Alternative to the Zassenhaus Formula

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Behr, N.; Dattoli, G.; Lattanzi, A.
Operator Ordering and Solution of Pseudo-Evolutionary Equations. *Axioms* **2019**, *8*, 35.
https://doi.org/10.3390/axioms8010035

**AMA Style**

Behr N, Dattoli G, Lattanzi A.
Operator Ordering and Solution of Pseudo-Evolutionary Equations. *Axioms*. 2019; 8(1):35.
https://doi.org/10.3390/axioms8010035

**Chicago/Turabian Style**

Behr, Nicolas, Giuseppe Dattoli, and Ambra Lattanzi.
2019. "Operator Ordering and Solution of Pseudo-Evolutionary Equations" *Axioms* 8, no. 1: 35.
https://doi.org/10.3390/axioms8010035