# Discrete Quantum Harmonic Oscillator

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Sequence of Discrete Quantum Harmonic Oscillators

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hounkonnou, M.N.; Arjika, S.; Baloitcha, E. Pöschl-Teller Hamiltonian: Gazeau-Klauder type coherent states, related statistics and geometry. J. Math. Phys.
**2014**, 55, 123502. [Google Scholar] [CrossRef] - Lorente, M. Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom. Phys. Lett. A
**2001**, 285, 119–126. [Google Scholar] [CrossRef] [Green Version] - Novikov, S.P.; Veselov, P. Exactly solvable two-dimensional Schrödinger operators and Laplace transformations. In Solitons, Geometry, and Topology: On the Crossroad; American Mathematical Society Translations: Series 2; American Mathematical Society: Providence, RI, USA, 1997; Volume 179, pp. 109–132. [Google Scholar]
- Boykin, T.B.; Klimeck, G. The discretized Schrödinger equation and simple models for semiconductor quantum wells. Eur. J. Phys.
**2004**, 25, 503–514. [Google Scholar] [CrossRef] - Bruckstein, A.M.; Kailath, T. On discrete Schrödinger equations and their two-component wave equation equivalents. J. Math. Phys.
**1987**, 28, 2914–2924. [Google Scholar] [CrossRef] - Fernández, D.J.; Fernández-García, N. Higher-order supersymmetric quantum mechanics. AIP Conf. Proc.
**2005**, 744, 236–273. [Google Scholar] - Fernández, D.J. Trends in supersymmetric quantum mechanics. In Integrability, Supersymmetry and Coherent States; CRM Series in Mathematical Physics; Springer: Cham, Switzerland, 2019; pp. 37–68. [Google Scholar]
- Goliński, T. Factorization method on time scales. Appl. Math. Comput.
**2019**, 347, 354–359. [Google Scholar] [CrossRef] - Tarasov, V.E. Exact discretization of Schrödinger equation. Phys. Lett. A
**2016**, 380, 68–75. [Google Scholar] [CrossRef] - Dobrogowska, A.; Fernández, D.J. Darboux transformations and second order difference equations. arXiv
**2018**, arXiv:1807.06895. [Google Scholar] - Elaydi, S. An Introduction to Difference Equations; Springer: New York, NY, USA, 1999. [Google Scholar]
- Álvarez-Nodarse, R.; Atakishiyev, N.M.; Costas-Santos, R.S. Factorization of the hypergeometric-type difference equation on the uniform lattice. ETNA Electron. Trans. Numer. Anal.
**2007**, 27, 34–50. [Google Scholar] - Nikiforov, A.F.; Suslov, S.K.; Uvarov, V.B. Classical orthogonal polynomials of a discrete variable. In Springer Series in Computational Physices; Springer: Berlin, Germany, 1991. [Google Scholar]
- Bavinck, H.; Vanhaeringen, H. Difference Equations for Generalized Meixner Polynomials. J. Math. An. App.
**1994**, 184, 453–463. [Google Scholar] [CrossRef] [Green Version] - Kruchinin, V.D.; Shablya, Y.V. Explicit Formulas for Meixner Polynomials. Int. J. Math. Math. Sci.
**2015**, 2015, 620569. [Google Scholar] [CrossRef] - Dobrogowska, A. New classes of second order difference equations solvable by factorization method. Appl. Math. Lett.
**2019**, 98, 300–305. [Google Scholar] [CrossRef] - Dobrogowska, A.; Hounkonnou, M.N. Factorization Method and General Second Order Linear Difference Equation. In Differential and Difference Equations with Applications; Pinelas, S., Caraballo, T., Kloeden, P., Graef, J., Eds.; ICDDEA 2017, Springer Proceedings in Mathematics & Statistics vol 230; Springer: Cham, Switzerland, 2018; pp. 67–77. [Google Scholar] [Green Version]
- Dobrogowska, A.; Jakimowicz, G. Factorization method applied to the second order difference equations. Appl. Math. Lett.
**2017**, 74, 161–166. [Google Scholar] [CrossRef] - Dobrogowska, A.; Odzijewicz, A. Second order q-difference equations solvable by factorization method. J. Comput. Appl. Math.
**2006**, 193, 319–346. [Google Scholar] [CrossRef] [Green Version] - Dobrogowska, A.; Odzijewicz, A. Solutions of the q-deformed Schrödinger equation for special potentials. J. Phys. A Math. Theor.
**2007**, 40, 2023–2036. [Google Scholar] [CrossRef] - Goliński, T.; Odzijewicz, A. Factorization method for second order functional equations. J. Comput. Appl. Math.
**2005**, 176, 331–355. [Google Scholar] [CrossRef] [Green Version] - Schrödinger, E. A method of determining quantum-mechanical eigenvalues and eigenfunctions. Proc. R. Ir. Acad. Sect. A
**1940**, 46, 9–16. [Google Scholar] - Infeld, L.; Hull, T.E. The Factorization Method. Rev. Mod. Phys.
**1951**, 23, 21–68. [Google Scholar] [CrossRef] - de Lange, O.L.; Raab, R.E. Operator Methods in Quantum Mechanics; Claredon Press: Oxford, UK, 1991. [Google Scholar]
- Mielnik, B. Factorization method and new potentials with the oscillator spectrum. J. Math. Phys.
**1984**, 25, 3387. [Google Scholar] [CrossRef] - Koekoek, R.; Lesky, P.A.; Swarttouw, R.F. Hypergeometric Orthogonal Polynomials and Their q-Analogues; Springer: Berlin/Heidelberg, Germany, 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**

Dobrogowska, A.; Fernández C., D.J.
Discrete Quantum Harmonic Oscillator. *Symmetry* **2019**, *11*, 1362.
https://doi.org/10.3390/sym11111362

**AMA Style**

Dobrogowska A, Fernández C. DJ.
Discrete Quantum Harmonic Oscillator. *Symmetry*. 2019; 11(11):1362.
https://doi.org/10.3390/sym11111362

**Chicago/Turabian Style**

Dobrogowska, Alina, and David J. Fernández C.
2019. "Discrete Quantum Harmonic Oscillator" *Symmetry* 11, no. 11: 1362.
https://doi.org/10.3390/sym11111362