# 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

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

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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