# A Fully Pseudo-Bosonic Swanson Model

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Assumption $\mathcal{D}$-pb 1.—**there exists a non-zero ${\phi}_{0}\in \mathcal{D}$, such that $a\phantom{\rule{0.166667em}{0ex}}{\phi}_{0}=0$.**Assumption $\mathcal{D}$-pb 2.—**there exists a non-zero ${\mathrm{\Psi}}_{0}\in \mathcal{D}$, such that ${b}^{\u2020}\phantom{\rule{0.166667em}{0ex}}{\mathrm{\Psi}}_{0}=0$.

**Assumption $\mathcal{D}$-pb 3.—**${\mathcal{F}}_{\phi}$ is a basis for $\mathcal{H}$.

**Assumption $\mathcal{D}$-pbw 3.—**${\mathcal{F}}_{\phi}$ and ${\mathcal{F}}_{\mathrm{\Psi}}$ are $\mathcal{G}$-quasi bases, for some subspace $\mathcal{G}$ dense (Notice that $\mathcal{G}$ does not need to coincide with $\mathcal{D}$, even if sometimes this happens.) in $\mathcal{H}$.

- $\forall f\in \mathcal{H}$. Looking at these expansions, it is natural to ask if sums, such as ${S}_{\phi}f={\sum}_{n=0}^{\infty}\u2329{\phi}_{n},f\u232a\phantom{\rule{0.166667em}{0ex}}{\phi}_{n}$ or ${S}_{\mathrm{\Psi}}f={\sum}_{n=0}^{\infty}\u2329{\mathrm{\Psi}}_{n},f\u232a\phantom{\rule{0.166667em}{0ex}}{\mathrm{\Psi}}_{n}$ also make some sense, or for which vectors they do converge, if any. In our case, since ${\mathcal{F}}_{\phi}$ and ${\mathcal{F}}_{\mathrm{\Psi}}$ are Riesz bases, we know that an orthonormal basis ${\mathcal{F}}_{e}=\left\{{e}_{n}\right\}$ exists, together with a bounded operator R with bounded inverse, such that ${\phi}_{n}=R{e}_{n}$ and ${\mathrm{\Psi}}_{n}={\left({R}^{-1}\right)}^{\u2020}{e}_{n}$, $\forall n$. It is clear that, if $R=1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1$, all these sums collapse and converge to f. However, what if $R\ne 1\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1$?

#### Leaving ${\mathcal{L}}^{2}\left(\mathbb{R}\right)$

**Proposition**

**1.**

## 3. The Model

## 4. Bi-Coherent States

**Theorem**

**1.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bender, C.M.; Fring, A.; Guenther, U.; Jones, H. Special issue on quantum physics with non-Hermitian operators. J. Phys. Math. Theor.
**2012**, 45, 030201. [Google Scholar] [CrossRef] - Bagarello, F.; Gazeau, J.P.; Szafraniec, F.H.; Znojil, M. (Eds.) Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects; John Wiley and Sons: Hoboken, NJ, USA, 2015. [Google Scholar]
- Bagarello, F.; Passante, R.; Trapani, C. Non-Hermitian Hamiltonians in Quantum Physics. In Proceedings of the 15th International Conference on Non-Hermitian Hamiltonians in Quantum Physics, Palermo, Italy, 18–23 May 2015; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Bender, C.M. PT Symmetry in Quantum and Classical Physics; World Scientific: Singapore, 2019. [Google Scholar]
- Bender, C.M.; Fring, A.; Correa, F. Proceedings for “Pseudo-Hermitian Hamiltonians in Quantum Physics”. J. Phys. Conf. Ser.
**2021**, 2038, 012001. [Google Scholar] - Bagarello, F. Deformed canonical (anti-)commutation relations and non hermitian Hamiltonians. In Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects; John Wiley and Sons: Hoboken, NJ, USA, 2015; pp. 121–188. [Google Scholar]
- Bagarello, F. Pseudo-Bosons and Their Coherent States; Springer: Berlin/Heidelberg, Germany, 2022. [Google Scholar]
- Trifonov, D.A. Pseudo-Boson Coherent and Fock States. In Proceedings of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia, Bulgaria, 25–29 August 2008; pp. 241–250. [Google Scholar]
- Jones, H.F. On pseudo-hermitian hamiltonians and their hermitian counterparts. J. Phys. A
**2005**, 38, 1741. [Google Scholar] [CrossRef] - Jones, H.F. The metric operator and the functional integral formulation of pseudo-hermitian quantum mechanics. Acta Polytech.
**2007**, 47, 36–39. [Google Scholar] [CrossRef] - Tavassoly, M.K. New nonlinear coherent states associated with inverse bosonic and f-deformed ladder operators. J. Phys. A
**2008**, 41, 285305. [Google Scholar] [CrossRef] - Bender, C.M.; Mannheim, P.D. Exactly solvable PT -symmetric Hamiltonian having no Hermitian counterpart. Phys. Rev. D
**2008**, 78, 025022. [Google Scholar] [CrossRef][Green Version] - da Providencia, J.; Bebiano, N.; da Providencia, J.P. Non hermitian operators with real spectrum in quantum mechanics. arXiv
**2010**, arXiv:0909.3054. [Google Scholar] - Swanson, M.S. Transition elements for a non-Hermitian quadratic hamiltonian. J. Math. Phys.
**2004**, 45, 585. [Google Scholar] [CrossRef] - Bagarello, F. Examples of Pseudo-bosons in quantum mechanics. Phys. Lett. A
**2010**, 374, 3823–3827. [Google Scholar] [CrossRef][Green Version] - Sinha, A.; Roy, P. Generalized Swanson Models and their solutions. J. Phys. A Math. Theor.
**2007**, 40, 10599. [Google Scholar] [CrossRef][Green Version] - Sinha, A.; Roy, P. Continuum states in generalized Swanson models. J. Phys. A Math. Theor.
**2009**, 42, 052002. [Google Scholar] [CrossRef][Green Version] - Graefe, E.-M.; Korsch, H.J.; Rush, A.; Schubert, R. Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator. J. Phys. A Math. Theor.
**2015**, 48, 055301. [Google Scholar] [CrossRef] - Fring, A.; Moussa, M.H.Y. The non-Hermitian Swanson model with a time-dependent metric. Phys. Rev. A
**2016**, 94, 042128. [Google Scholar] [CrossRef][Green Version] - Bagarello, F. A Swanson-like Hamiltonian and the inverted harmonic oscillator. J. Phys. A
**2022**, 55, 225204. [Google Scholar] [CrossRef] - Christensen, O. An Introduction to Frames and Riesz Bases; Birkhäuser: Boston, MA, USA, 2003. [Google Scholar]
- Young, R.M. On complete biorthogonal bases. Proc. Am. Math. Soc.
**1981**, 83, 537–540. [Google Scholar] [CrossRef] - Yeşiltaş, O.; Kaplan, N. Metric Operator For The Non-Hermitian Hamiltonian Model and Pseudo-Supersymmetry. Phys. Scr.
**2013**, 87, 045013. [Google Scholar] [CrossRef][Green Version] - Mostafazadeh, A. Metric Operators for Quasi-Hermitian Hamiltonians and Symmetries of Equivalent Hermitian Hamiltonians. J. Phys. A Math. Theor.
**2008**, 41, 055304. [Google Scholar] [CrossRef][Green Version] - Bagarello, F. A class of weak pseudo-bosons and their bi-coherent states. JMAA
**2022**, 516, 126531. [Google Scholar] [CrossRef] - Bagarello, F. Weak pseudo-bosons. J. Phys. A
**2020**, 53, 135201. [Google Scholar] [CrossRef][Green Version] - Bagarello, F. Pseudo-bosons and bi-coherent states out of L
^{2}(**R**). J. Phys. Conf. Ser.**2021**, 2038, 012001. [Google Scholar] [CrossRef] - Szegö, G. Orthogonal Polynomials; AMS: Ann Arbor, MI, USA, 1939. [Google Scholar]
- Davies, E.B.; Kuijlaars, B.J. Spectral asymptotics of the non-self-adjoint harmonic oscillator. J. London Math. Soc.
**2004**, 70, 420–426. [Google Scholar] [CrossRef] - Kolmogorov, A.; Fomine, S. Eléments de la Théorie des Fonctions et de l’Analyse Fonctionnelle; Mir: Moscow, Russia, 1973. [Google Scholar]
- Merzbacher, E. Quantum Mechanics; Wiley: New York, NY, USA, 1970. [Google Scholar]
- Messiah, A. Quantum Mechanics; North Holland Publishing Company: Amsterdam, The Netherlands, 1962; Volume 2. [Google Scholar]
- Bagarello, F.; Gargano, F.; Spagnolo, S. Two-dimensional non commutative Swanson model and its bicoherent states. In Geometric Methods in Physics, XXXVI; Kielanowski, P., Odzijewicz, A., Previato, E., Eds.; Trends in Mathematics; Birkhäuser: Basel, Switzerland, 2019; pp. 9–19. [Google Scholar]
- Bagarello, F.; Feinberg, J. Bicoherent-state path integral quantization of a non-Hermitian Hamiltonian. Ann. Phys.
**2020**, 422, 168313. [Google Scholar] [CrossRef]

**Figure 1.**${\left|\psi (z;x)\right|}^{2}$ (orangish) and ${\left|\phi (z;x)\right|}^{2}$ (blueish) in (50) and (51) for different $\alpha $ and $\beta $ and for $\lambda =0.1$ and $\omega =0.5$: (

**top left**) $\alpha =0.3$, $\beta =0.31$; (

**top right**) $\alpha =0.3$, $\beta =0.35$; (

**bottom left**) $\alpha =0.3$, $\beta =0.5$; (

**bottom right**) $\alpha =0.3$, $\beta =1$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bagarello, F. A Fully Pseudo-Bosonic Swanson Model. *Mathematics* **2022**, *10*, 3954.
https://doi.org/10.3390/math10213954

**AMA Style**

Bagarello F. A Fully Pseudo-Bosonic Swanson Model. *Mathematics*. 2022; 10(21):3954.
https://doi.org/10.3390/math10213954

**Chicago/Turabian Style**

Bagarello, Fabio. 2022. "A Fully Pseudo-Bosonic Swanson Model" *Mathematics* 10, no. 21: 3954.
https://doi.org/10.3390/math10213954