# Operator Ordering and Solution of Pseudo-Evolutionary Equations

^{1}

^{2}

^{3}

^{*}

## 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)$${}_{l}e(x)=\widehat{\mathbb{I}}\left(v{e}^{vx}\right)\phantom{\rule{0.166667em}{0ex}}.$$Here, v is a formal variable, and $\widehat{\mathbb{I}}$ a formal integration operator, which acts according to$$\widehat{\mathbb{I}}\left({v}^{\alpha}\right):=\frac{1}{\mathsf{\Gamma}(\alpha )}\phantom{\rule{2.em}{0ex}}(\alpha \in \mathbb{C})\phantom{\rule{0.166667em}{0ex}}.$$

- 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$$F(x,t)=\widehat{\mathbb{I}}\left(v{e}^{-\frac{{(vt)}^{2}}{2}\alpha \beta}{e}^{-vt\alpha \widehat{x}}{e}^{vt\beta {\partial}_{x}}f(x)\right)=\widehat{\mathbb{I}}\left(v{e}^{-\frac{{(vt)}^{2}}{2}\alpha \beta}{e}^{-vt\alpha \widehat{x}}f(x+v\beta t)\right)\phantom{\rule{0.166667em}{0ex}}.$$

## 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

## References

- Babusci, D.; Dattoli, G. Umbral methods and operator ordering. arXiv, 2011; arXiv:1112.1570. [Google Scholar]
- Dattoli, G.; Gorska, K.; Horzela, A.; Licciardi, S.; Pidatella, R.M. Comments on the properties of Mittag-Leffler function. Eur. Phys. J. Spec. Top.
**2017**, 226, 3427–3443. [Google Scholar] [CrossRef] - Dattoli, G. Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle. In Proceedings of the Workshop on Advanced Special Functions and Applications; Aracne: Rome, Italy, 1999. [Google Scholar]
- Weyl, H. The Theory of Groups and Quantum Mechanics; Courier Corporation: North Chelmsford, MA, USA, 1950. [Google Scholar]
- Wilcox, R.M. Exponential operators and parameter differentiation in quantum physics. J. Math. Phys.
**1967**, 8, 962–982. [Google Scholar] [CrossRef] - Dattoli, G.; Gallardo, J.C.; Torre, A. An algebraic view to the operatorial ordering and its applications to optics. Riv. Nuovo Cimento Ser. 3
**1988**, 11, 1–79. [Google Scholar] [CrossRef] - Casas, F.; Murua, A.; Nadinic, M. Efficient computation of the Zassenhaus formula. Comput. Phys. Commun.
**2012**, 183, 2386–2391. [Google Scholar] [CrossRef] - Gill, T.L. The Feynman-Dyson view. J. Phys. Conf. Ser.
**2017**, 845, 012023. [Google Scholar] [CrossRef] - Ditkin, V.A.; Prudnikov, A.P. Operational calculus. Itogi Nauk. Tekhnik. Ser. Mat. Anal.
**1967**, 4, 7–82. [Google Scholar] - Penson, K.A.; Blasiak, P.; Horzela, A.; Duchamp, G.H.E.; Solomon, A.I. Laguerre-type derivatives: Dobiński relations and combinatorial identities. J. Math. Phys.
**2009**, 50, 083512. [Google Scholar] [CrossRef] - Dattoli, G. Generalized polynomials, operational identities and their applications. J. Comput. Appl. Math.
**2000**, 118, 111–123. [Google Scholar] [CrossRef] - Dattoli, G.; Srivastava, H.M.; Zhukovsky, K. A new family of integral transforms and their applications. Integr. Trans. Spec. Funct.
**2006**, 17, 31–37. [Google Scholar] [CrossRef] - Dattoli, G.; He, M.; Ricci, P. Eigenfunctions of laguerre-type operators and generalized evolution problems. Math. Comput. Modell.
**2005**, 42, 1263–1268. [Google Scholar] [CrossRef] - Dattoli, G.; Licciardi, S.; Pidatella, R. Theory of generalized trigonometric functions: From Laguerre to Airy forms. J. Math. Anal. Appl.
**2018**, 468, 103–115. [Google Scholar] [CrossRef] - Babusci, D.; Dattoli, G.; Górska, K.; Penson, K. The spherical Bessel and Struve functions and operational methods. Appl. Math. Comput.
**2014**, 238, 1–6. [Google Scholar] [CrossRef] - Behr, N.; Dattoli, G.; Duchamp, G.H.; Penson, S. Operational methods in the study of Sobolev-Jacobi polynomials. Mathematics
**2019**, 7, 124. [Google Scholar] [CrossRef] - Crofton, M.W. Theorems in the calculus of operations. Q. J. Math.
**1879**, 16, 323–352. [Google Scholar] - Dattoli, G.; Ottaviani, P.L.; Torre, A.; Vázquez, L. Evolution operator equations: Integration with algebraic and finitedifference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory. Riv. Nuovo Cimento
**1997**, 20, 3–133. [Google Scholar] [CrossRef] - Mittag-Leffler, G. Sur la nouvelle fonction Eα (x). CR Acad. Sci. Paris
**1903**, 137, 554–558. [Google Scholar] - Diethelm, K. The Analysis of Fractional Differential Equations; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar] [CrossRef]
- Louisell, W.H. Quantum Statistical Properties of Radiation; Wiley: New York, NY, USA, 1973; Volume 7. [Google Scholar]
- Berry, M.V. The Diffraction of Light by Ultrasound; Academic Press: Cambridge, MA, USA, 1966. [Google Scholar]
- Babusci, D.; Dattoli, G.; Del Franco, M. Lectures on Mathematical Methods for Physics; Thecnical Report; ENEA: Rome, Italy, 2010. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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