# Generalized para-Bose states

A. S. Pereira<EMAIL_ADDRESS>Instituto Federal do Pará, 68600-000
Bragança, Pará, Brazil  Departamento de Física, Universidade Federal de
Campina Grande, Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil
A. S. Lemos<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
de Campina Grande, Caixa Postal 10071, 58429-900 Campina Grande, Paraíba,
Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande, Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil

###### Abstract

In this paper, we construct integrals of motion in a para-Bose formulation for
a general time-dependent quadratic Hamiltonian, which, in its turn, commutes
with the reflection operator. In this context, we obtain generalizations for
the squeezed vacuum states (SVS) and coherent states (CS) in terms of the
Wigner parameter. Furthermore, we show that there is a completeness relation
for the generalized SVS owing to the Wigner parameter. In the study of the
probability transition, we found that the displacement parameter acts as a
transition parameter by allowing access to odd states, while the Wigner
parameter controls the dispersion of the distribution. We show that the Wigner
parameter is quantized by imposing that the vacuum state has even parity. We
apply the general results to the case of the time-independent para-Bose
oscillator and find that the mean values of the coordinate and momentum have
an oscillatory behavior similarly to the simple harmonic oscillator, while the
standard deviation presents corrections in terms of the squeeze, displacement,
and Wigner parameters.

Coherent states, squeezed states, integrals of motion, Wigner-Heisenberg
algebra.

###### pacs:

03.65.Sq, 03.65.Fd, 03.65.Ca

##  I Introduction

In 1926 Schrödinger obtained quantum states to harmonic oscillator, which
allowed for a semiclassical description (SCH1926, ). In the 1960s, with the
advent of the laser, these states were rediscovered and widespread in the
works of Glauber-Sudarshan-Klauber (GLA1963, ; SUD1963, ; KLA1963, ) by
showing a semiclassical description of electromagnetic radiation, and where
the term “coherent states” (CS) was presented for the first time. These states
for the harmonic oscillator can be obtained in three equivalent ways: I - as
eigenstates of the annihilation operator; II - by the action of a unitary
displacement operator acting in the vacuum state; III - as minimum uncertainty
states. The minimization of the uncertainty relation coincides with the value
calculated in the vacuum state of the harmonic oscillator, which presents the
same value for the standard deviation in both position and momentum. The
states satisfying these properties are called canonical CS. In addition, these
states have properties of continuity and form an overcomplete basis, which
guarantees a prominent role in modern quantum mechanics, with various
applications ranging from quantum optics (SCU1997, ; KLA1968, ), quantum
computing (NIE2000, ), and mathematical physics (KLA1985, ).

In its turn, squeezed states (SS) form a class of nonclassical states, which
minimize the uncertainty relation Bac2004 ; SCH2017 ; SAN2021 . At the same
time, these states offer the possibility of reducing the standard deviation of
a physical quantity to a value less than that calculated in the vacuum state
at the expense of increasing the standard deviation of other physical quantity
Sto1970 ; Sto1971 . Thus, these states have great potential for application,
for example, in quantum information Slu1990  and for the detectors of
gravitational waves Cav1981 ; Ni1987 ; Chu2014 . Usually, these states are
constructed by the action of the unitary squeeze operator on an arbitrary
state. In particular, the acting of the squeeze operator on the vacuum state
leads to the squeezed vacuum states (SVS) Ger1983 .

Generalizing quantum states provides a deeper understanding of corresponding
systems since they sometimes bring additional degrees of freedom. In this
sense, there is a wide range of publications in the literature that seeks to
generalize CS to be able to describe systems in addition to the harmonic
oscillator, such as the time-dependent quadratic systems (MAL1979, ; DOD2003,
), systems with a given Lie group (PER1986, ; GIL1972, ), some non-trivial
generalizations (ALI2000, ), and squeezed coherent states (SCS) (YUE1976, ;
SAT1985, ). On the other hand, CS can be generalized by considering
deformations in the canonical commutation relation and thus allow us to study
a wide range of relevant problems. For instance, the implications of the
gravitational effects in quantum mechanics (DEY2012, ; PED2013, ; PAT2021, ),
q- and f-deformed oscillators study which is applied, mainly, to quantum
optics (MAN1997, ; SAN2015, ; ERE2006, ), and to study systems with
singularity as the Calogero-like model via Wigner-Heisenberg algebra (WHA) (or
R-deformed Heisenberg algebra) Aga1995 ; BRI1993 . In particular, we are
interested in generalizing CS and SVS via WHA.

The WHA originated from the quantization proposed by Wigner (WIG1950, ), which
generalizes the canonical commutation relation. In particular, this algebra is
generated by the creation-annihilation operators and the reflection operator
(parity operator), which satisfies commutation and anti-commutation relations
(MIK1997, ). In this formulation, it is possible to obtain a self-adjoint
momentum operator on the semi-axis in terms of a parameter that deforms the
canonical commutation relation MUK1980 ; LOH2004 . Furthermore, the WHA obeys
the trilinear commutation and anti-commutation relations, which characterize
the para-Bose operators SHA1981 . On its turn, this algebra leads to the
parastatistical description of physical systems in terms of a deformation
parameter, which corresponds to generalization of the Bose-Einstein and Fermi-
Dirac statistics (HAR1968, ; GRE1953, ; OHN1982, ).

According to the order of the deformation parameter, also known as the order
of statistics, it is possible to describe para-Bose or para-Fermi particles,
see also (STO2010, ). Although parastatistics cannot be applied to the
particles described by the standard model (GRE1965, ), their distinctive
approach has allowed presenting promising proposals, for instance, in the
conjecture of candidate particles to explain dark matter (KIT2018, ), in
paraquark models description (BRA1973, ), in the study of thermodynamics
properties of para-Bose systems (HAM1992, ), and in optical physics by
simulation of para-Fermi oscillators (WAL2020, ). By considering the para-Bose
formulation, we will construct integrals of motion, i.e., operators that
commute with the Schrödinger’s operator (PER2021, ) in the form of a
Bogoliubov transformation (MEY1980, ).

In the para-Bose formulation (JOR1963, ), the CS were obtained as eigenstates
of the annihilation operator that satisfy the characteristic commutation
relation of the algebra (SHA1978, ) – see (SHA1981, ; BIS1980, ; SAX1986, ).
Recently, some works have paid attention to this topic, for instance, the
nonlinear CS (MOJ2018a, ), the construction of new types of para-Bose states
MOJ2018b ; Ald2017 ; Bag1997 , and the study of “Schrödinger cat states”
(DEH2015, ). In this context, we will obtain the time-dependent generalized CS
and SVS via integrals of motion. These states can be applied to describe
Calogero-like models CAL1971 . In its turn, the Calogero model and its
generalizations are able to describe several physical phenomena SIM1994 , such
as the Hall effect KAW1993 ; AZU1994 , anyons OUV2001 , and fluctuations in
mesoscopic systems CAS1995 , for instance. Furthermore, our construction has
potential application to describe the recent proposed experimental realization
of para-particles ALD2021 ; ALD2017 .

This paper is structured as follows. In Sec. II, the integrals of motion
method in the context of WHA will be formulated. In Sec. III, the time-
dependent generalized SVS in terms of the time-independent para-Bose number
states will be constructed. In turn, the completeness relation and probability
transition plots will be obtained. In Sec. IV, a generalization of the time-
dependent CS in terms of the Wigner, squeeze and displacement parameters will
be constructed. Then, the probability transition graph will be shown, and the
mean values and the uncertainty relations will be calculated. In Sec. V, the
generalized CS in the coordinate representation will be considered. We apply
the general result to the particular case of the para-Bose oscillator in Sec.
VI. Then, the concluding remarks are presented in Sec. VII.

##  II Integrals of motion via WHA

The motion integral method consists of building a time-dependent operator,
which commutes with the Schrödinger’s operator. In turn, the eigenstates of
this operator are obtained and imposed to satisfy the Schrödinger’s equation.
This technique introduced by Lewis and Reisenfield to study the time-dependent
harmonic oscillator Lew1969  proved to be useful in several other problems,
e.g., in the study of time-dependent quadratic Hamiltonians in one dimension
DOD1975 ; Bag2015 ; And1999  and for multidimensional systems Dod1989 , Dirac
equation Man1997 , and supersymmetric quantum mechanics Mos2001 . Here, our
contribution will be to reformulate the integrals of motion method according
to the WHA. This algebraic formulation allows studying time-dependent systems
with a singularity at the origin characterized by centrifugal potentials.

The WHA is composed by the annihilation $\hat{a}$, creation
$\hat{a}^{\dagger}$ and reflection $\hat{R}$ operators, satisfying commutation
$\left(\left[\hat{b},\hat{c}\right]=\hat{b}\hat{c}-\hat{c}\hat{b}\right)$ and
anti-commutation
$\left(\left\\{\hat{b},\hat{c}\right\\}=\hat{b}\hat{c}+\hat{c}\hat{b}\right)$
relations,

$\left[\hat{a},\hat{a}^{\dagger}\right]=1+\nu\hat{R},\text{ \
}\left\\{\hat{R},\hat{a}\right\\}=0=\left\\{\hat{R},\hat{a}^{\dagger}\right\\},\text{
\ }\hat{R}^{2}=1,$ (1)

where $\nu$ is the Wigner parameter (deformation parameter) related to the
fundamental energy level $\varepsilon$ of the para-Bose oscillator given by

$\nu=2\varepsilon-1.$ (2)

Notice that by setting $\nu=0\Rightarrow\varepsilon=1/2$ we recover the
canonical commutation relation.

The operators $\hat{a}$ and $\hat{a}^{\dagger}$ also obey the trilinear
commutation and anti-commutation relation

$\left[\left\\{\hat{a},\hat{a}^{\dagger}\right\\},\hat{a}\right]=-2\hat{a},\text{
\
}\left[\left\\{\hat{a},\hat{a}^{\dagger}\right\\},\hat{a}^{\dagger}\right]=2\hat{a}^{\dagger},$
(3)

which characterize the para-Bose operators MIK1997 ; SHA1981 .

For the sake of simplicity, we will consider a Hamiltonian which commutes with
the reflection operator, and in that way, the eigenstates of $\hat{H}$ can be
even or odd. In this sense, the most general form for a one-dimensional time-
dependent quadratic Hamiltonian is given by

$\hat{H}=\frac{1}{2}\hbar\left(\alpha^{\ast}\hat{a}^{2}+\alpha\hat{a}^{\dagger
2}\right)+\frac{1}{2}\hbar\beta\left(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}\right)+\hbar\delta,$
(4)

where $\alpha=\alpha\left(t\right)$, $\beta=\beta\left(t\right)$ and
$\delta=\delta\left(t\right)$ are time-dependent functions and the signs
$\dagger$ and $\ast$ denote Hermitian and complex conjugation, respectively.
From the hermiticity condition $\hat{H}=\hat{H}^{\dagger}$, we have that
$\beta$ and $\delta$ must be real functions. By assuming the condition
$\beta>\left|\alpha\right|$, we get that the Hamiltonian (4) is positive
definite, i.e., it can be written in the form of an oscillator-type
Hamiltonian.

In its turn, the quantum states $\left|\Psi\right\rangle$ which describe the
time evolution of the system should satisfy the Schrödinger’s equation

$\displaystyle\hat{\Lambda}\left|\Psi\right\rangle=0,\text{ \ }$
$\displaystyle\hat{\Lambda}=\frac{i}{\hbar}\hat{H}+\partial_{t},\text{ \
}\partial_{t}=\frac{\partial}{\partial t},$ (5)

where $\hat{\Lambda}$ is the Schrödinger’s operator.

Let us consider a time-dependent operator $\hat{A}=\hat{A}\left(t\right)$, as
a linear combination of the $\hat{a}$ and $\hat{a}^{\dagger}$ operators, in
the form

$\hat{A}=f\hat{a}+g\hat{a}^{\dagger}+\varphi,$ (6)

where $f=f\left(t\right)$, $g=g\left(t\right)$ and
$\varphi=\varphi\left(t\right)$ are time-dependent complex functions. The Eq.
(6) is the well-known Bogoliubov’s transformation; those coefficients have
been studied in Ref. (MEY1980, ).

In order to $\hat{A}$ to be an integral of motion, it has to commute with the
Schrödinger’s operator (5), that is

$\overset{\cdot}{\hat{A}}=\left[\hat{\Lambda},\hat{A}\right]=0,\text{ \
}\overset{\cdot}{\hat{A}}=\frac{d\hat{A}}{dt}.$ (7)

Substituting the Eqs. (4), (5) and (6) into (7), we obtain the following
equations for the functions $f$, $g$ and $\varphi$:

$\dot{f}=i\left(\beta f-\alpha^{\ast}g\right),\text{ \
}\dot{g}=i\left(f\alpha-\beta g\right),\text{ \ }\dot{\varphi}=0,$ (8)

where $\varphi\left(t\right)=\varphi_{0}$ is a constant function for any
instant of time.

The commutator between $\hat{A}$ and $\hat{A}^{\dagger}$ reads

$\left[\hat{A},\hat{A}^{\dagger}\right]=\mu\left(1+\nu\hat{R}\right),\text{\
}\mu=\left|f\right|^{2}-\left|g\right|^{2}=\left|f_{0}\right|^{2}-\left|g_{0}\right|^{2},\text{\
}\text{for arbitrary $t$},$ (9)

where $f_{0}=f\left(0\right)$ and $g_{0}=g\left(0\right)$ are initial
conditions.

It follows from (6) and (9), that

$\hat{a}=\frac{f^{\ast}\hat{A}-g\hat{A}^{\dagger}+u}{\mu},\text{\
}u=g\varphi_{0}^{\ast}-f^{\ast}\varphi_{0}.$ (10)

Notice that $\left|f\right|\neq\left|g\right|$ for all time ensures that
$\hat{a}$ is well defined for all $t$. It is worth highlighting that the
$\mu$-parameter will be useful to study special cases in which the Hamiltonian
presents linear terms. In addition, the $\mu$-parameter will be useful to
replace the functions $f$ and $g$ with the squeeze and displacement parameters
in the constructed states.

##  III Time-dependent generalized SVS

The SVS are pure states known to be one of the most useful nonclassical
states, see for instance, SCH2017 . In addition, these states make it possible
to obtain the standard deviation of a physical quantity less than its value in
the vacuum state. In this section, we will obtain the time-dependent
generalized SVS via para-Bose formulation. Thus, assuming the condition

$\varphi_{0}=0\Longrightarrow u=0,$ (11)

one can obtain the generalized SVS following the nonunitary approach, as
described in Ref. PER2021 . In what follows we will apply this condition.

###  III.1 Para-Bose number states

In this subsection, we will recall some properties of the para-Bose number
states, since the generalized SVS can be expanded in these states. It is well-
known that the para-Bose number states $\left|n,\varepsilon\right\rangle$ form
a complete set and are orthonormal (SHA1978, ; SHA1981, ), i.e.,

${\displaystyle\sum\limits_{n=0}^{\infty}}\left|n,\varepsilon\right\rangle\left\langle\varepsilon,n\right|=1,\text{
\ }\left\langle\varepsilon,n|m,\varepsilon\right\rangle=\delta_{n,m},$ (12)

where $\delta_{n,m}$ is the Kronecker delta.

Taking into account that the vacuum state has even parity
$\left(\hat{R}\left|0,\varepsilon\right\rangle=\left|0,\varepsilon\right\rangle\right)$,
the action of the generators of the WHA in $\left|n,\varepsilon\right\rangle$,
reads

$\displaystyle\hat{a}\left|2n,\varepsilon\right\rangle=\sqrt{2n}\left|2n-1,\varepsilon\right\rangle,\text{
\
}\hat{a}^{\dagger}\left|2n,\varepsilon\right\rangle=\sqrt{2\left(n+\varepsilon\right)}\left|2n+1,\varepsilon\right\rangle,$
$\displaystyle\hat{a}\left|2n+1,\varepsilon\right\rangle=\sqrt{2\left(n+\varepsilon\right)}\left|2n,\varepsilon\right\rangle,\text{
\
}\hat{a}^{\dagger}\left|2n+1,\varepsilon\right\rangle=\sqrt{2\left(n+1\right)}\left|2n+2,\varepsilon\right\rangle,$
$\displaystyle\hat{n}\left|n,\varepsilon\right\rangle=\left(\frac{1}{2}\left\\{\hat{a},\hat{a}^{\dagger}\right\\}-\varepsilon\right)\left|n,\varepsilon\right\rangle=n\left|n,\varepsilon\right\rangle,\text{
\
}\hat{R}\left|n,\varepsilon\right\rangle=\left(-1\right)^{n}\left|n,\varepsilon\right\rangle,\text{
\ }n=0,1,2,3,\ldots.$ (13)

From a recurrence relation, one can write the number states
$\left|n,\varepsilon\right\rangle$ in terms of the vacuum state
$\left|0,\varepsilon\right\rangle$, in the form

$\displaystyle\left|2n,\varepsilon\right\rangle=\sqrt{\frac{\Gamma\left(\varepsilon\right)}{2^{2n}n!\Gamma\left(n+\varepsilon\right)}}\left(\hat{a}^{\dagger}\right)^{2n}\left|0,\varepsilon\right\rangle,$
$\displaystyle\left|2n+1,\varepsilon\right\rangle=\sqrt{\frac{\Gamma\left(\varepsilon\right)}{2^{2n+1}n!\Gamma\left(n+\varepsilon+1\right)}}\left(\hat{a}^{\dagger}\right)^{2n+1}\left|0,\varepsilon\right\rangle.$
(14)

###  III.2 Time-dependent generalized SVS via para-Bose number states

In what follows, we aim to apply the nonunitary approach, as in the recent
publication (PER2021, ), to construct the generalized SVS. From condition
(11), one can write the nonunitary operator $\hat{S}$, in the form

$\hat{S}=\exp\left(\frac{1}{2}\zeta\hat{a}^{\dagger 2}\right),\text{ \
}\zeta=\frac{g}{f},\text{ \
}\dot{\zeta}=i\alpha^{\ast}\zeta^{2}-2i\beta\zeta+i\alpha,$ (15)

such that the commutators from Baker-Campbell-Hausdorff relation of the second
order onwards become null. Here, $\zeta=\zeta\left(t\right)$ represents the
squeeze parameter for the generalized SVS. The form of $\hat{S}$ allows us to
introduce the most convenient squeeze parameter, as well as relate the motion
integral $\hat{A}$ directly with the annihilation operator $\hat{a}$.
Furthermore, this approach guarantees that there is a direct relationship
between the eigenstates of the motion integral with the para-Bose number
states, as we will see next.

Applying the Baker–Campbell–Hausdorff theorem, we can express the operator
$\hat{a}$ in terms of the integrals of motion $\hat{A}$, as follows

$\hat{a}=\frac{1}{f}\hat{S}\hat{A}\hat{S}^{-1}.$ (16)

The application from (16) on the vacuum state
$\left|0,\varepsilon\right\rangle$, which satisfies the annihilation condition
$\hat{a}\left|0,\varepsilon\right\rangle=0$, yields:

$\hat{A}\left|\zeta\right\rangle=0,\text{ }$ (17)

whose general solution is given by

$\text{\
}\left|\zeta\right\rangle=\Phi\exp\left(-\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)\left|0,\varepsilon\right\rangle,$ (18)

where $\Phi=\Phi\left(t\right)$ is an arbitrary function, which will be
determined such that the states $\left|\zeta\right\rangle$ satisfy the
Schrödinger’s equation (5).

Substituting the states (18) into (5), we obtain the following equation for
$\Phi$:

$\frac{\dot{\Phi}}{\Phi}=\frac{1}{2}\frac{\left\langle\zeta\left|\hat{a}^{\dagger
2}\right|\zeta\right\rangle}{\left\langle\zeta|\zeta\right\rangle}\dot{\zeta}-\frac{i}{\hbar}\frac{\left\langle\zeta\left|\hat{H}\right|\zeta\right\rangle}{\left\langle\zeta|\zeta\right\rangle}.$
(19)

Using the representation (10) together with the condition (17), one can easily
calculate the mean values in (19), with $\dot{\zeta}$ given in (15). Thus, the
general solution from (19), reads

$\Phi=\frac{C}{f^{\varepsilon}}\exp\left(-i\int\delta dt\right),$ (20)

where $C$ is a real normalization constant. Taking into account the
normalization condition, we find that:

$\left\langle\zeta|\zeta\right\rangle=1\Rightarrow C=\mu^{\varepsilon/2}.$
(21)

Then, the normalized states $\left|\zeta\right\rangle$ that satisfy the
Schrödinger’s equation are given by

$\displaystyle\left|\zeta\right\rangle$
$\displaystyle=\frac{\sqrt{\mu^{\varepsilon}}}{f^{\varepsilon}}\exp\left(-i\int\delta
dt\right)\exp\left(-\frac{1}{2}\zeta\hat{a}^{\dagger
2}\right)\left|0,\varepsilon\right\rangle$
$\displaystyle=\frac{\sqrt{\mu^{\varepsilon}}}{f^{\varepsilon}}\exp\left(-i\int\delta
dt\right){\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta\right)^{n}\sqrt{\frac{\Gamma\left(n+\varepsilon\right)}{n!\Gamma\left(\varepsilon\right)}}\left|2n,\varepsilon\right\rangle.$
(22)

The above sum converges as long as the condition $\left|\zeta\right|<1$ is
satisfied.

On the other hand, we can express $f$ and $\mu$ in terms of squeeze parameter
in the form:

$f=f_{0}\exp\left[-i\int\left(\alpha^{\ast}\zeta-\beta\right)dt\right],\text{
\ }\mu=\left|f\right|^{2}\left(1-\left|\zeta\right|^{2}\right).$ (23)

From here, the states $\left|\zeta\right\rangle$ take the form

$\left|\zeta\right\rangle=\left(1-\left|\zeta\right|^{2}\right)^{\frac{\varepsilon}{2}}e^{i\vartheta}{\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta\right)^{n}\sqrt{\frac{\Gamma\left(n+\varepsilon\right)}{n!\Gamma\left(\varepsilon\right)}}\left|2n,\varepsilon\right\rangle,$
(24)

where the phase $\vartheta$ is given by

$\vartheta=\int\left[\varepsilon\operatorname{Re}\left(\alpha\zeta^{\ast}\right)-\varepsilon\beta-\delta\right]dt.$
(25)

In what follows, we call the time-dependent states (24) generalized SVS.
Notice that, if we assume $\varepsilon=1/2$ and
$\zeta=e^{i\theta}\tanh\left(r\right)\Longrightarrow\mu=1$, the Eq. (24) takes
the form:

$\left|\zeta\right\rangle=\frac{e^{i\vartheta}}{\sqrt{\cosh(r)}}\sum_{n=0}^{\infty}\frac{\sqrt{(2n)!}}{2^{n}n!}\left[-e^{i\theta}\tanh(r)\right]^{n}\left|2n,\frac{1}{2}\right\rangle,$
(26)

where $\Gamma\left(n+\frac{1}{2}\right)=\frac{\sqrt{\pi}4^{-n}(2n)!}{n!}$,
$\Gamma\left(1/2\right)=\sqrt{\pi}$. So, we can conclude that the states
obtained in Eq. (26) reproduce the usual case QUE2001 . In this case, the Eq.
(24) corresponds to a generalization of the SVS, differing by a time-dependent
phase factor due to the time evolution that was included in our analysis. We
must highlight that when identifying $\varepsilon=2k$, the states (24)
coincide with the time-independent CS of the $SU\left(1,1\right)$ group
constructed by Peremolov PER1986 .

The overlap of two generalized SVS with different $\zeta$, for example
$\left\langle\zeta_{1}|\zeta_{2}\right\rangle$ reads

$\left\langle\zeta_{1}|\zeta_{2}\right\rangle=\frac{\left(1-\left|\zeta_{1}\right|^{2}\right)^{\frac{\varepsilon}{2}}\left(1-\left|\zeta_{2}\right|^{2}\right)^{\frac{\varepsilon}{2}}}{\left(1-\zeta_{1}^{\ast}\zeta_{2}\right)^{\varepsilon}}\exp\left\\{i\varepsilon\int\operatorname{Re}\left[\alpha\left(\zeta_{2}^{\ast}-\zeta_{1}^{\ast}\right)\right]dt\right\\}.$
(27)

In turn, the probability transition
$P_{2n}\left(\zeta,\varepsilon\right)=\left|\left\langle\varepsilon,2n|\zeta\right\rangle\right|^{2}$
is given by

$P_{2n}\left(\zeta,\varepsilon\right)=\frac{\left(1-\left|\zeta\right|^{2}\right)^{\varepsilon}}{\Gamma\left(\varepsilon\right)}\frac{\Gamma\left(n+\varepsilon\right)\left|\zeta\right|^{2n}}{n!}.$
(28)

The probability transition (28) has been plotted in Fig. 1. As we see, the
increase in the value of the $\varepsilon$ parameter implies a larger
dispersion of the probability transition.

(a)  (b)  (c)  (d)

Figure 1: Probability transition of the generalized SVS by considering fixed
value for $\left|\zeta\right|=0.3$.

###  III.3 Completeness relation on squeeze parameter

Considerations on the completeness relation for the SVS and squeezed odd
number states, in the context of canonical algebra, can be seen in Refs.
(CHA1996, ; QUE2001, ). Here, our contribution is to obtain the completeness
relation in the context of the WHA, and as will be seen, its existence depends
on the $\varepsilon$-parameter.

Let us consider a weight function $w\left(\zeta\right)$ such that the states
$\left|\zeta\right\rangle$ lead to the following closure relation

$\int_{\mathbb{C}}\left|\zeta\right\rangle\left\langle\zeta\right|w\left(\zeta\right)d^{2}\zeta=1.$
(29)

In turn, substituting the states (24) into Eq. (29), we find

${\displaystyle\sum\limits_{n,m=0}^{\infty}}\left|2n,\varepsilon\right\rangle\left\langle\varepsilon,2m\right|\sqrt{\frac{\Gamma\left(n+\varepsilon\right)}{n!\Gamma\left(\varepsilon\right)}\frac{\Gamma\left(m+\varepsilon\right)}{m!\Gamma\left(\varepsilon\right)}}\int_{\mathbb{C}}\left(1-\left|\zeta\right|^{2}\right)^{\varepsilon}\left(-\zeta\right)^{n}\left(-\zeta^{\ast}\right)^{m}w\left(\zeta\right)d^{2}\zeta=1.$
(30)

Now, by using polar coordinates in the above relation,

$\zeta=r_{\zeta}e^{i\theta_{\zeta}},\text{ \
}d^{2}\zeta=r_{\zeta}dr_{\zeta}d\theta_{\zeta},\text{ \ }0\leq
r_{\zeta}<1,\text{ \ }0\leq\theta_{\zeta}\leq 2\pi,$ (31)

we get

$\displaystyle{\displaystyle\sum\limits_{n,m=0}^{\infty}}\left|2n,\varepsilon\right\rangle\left\langle\varepsilon,2m\right|\sqrt{\frac{\Gamma\left(n+\varepsilon\right)}{n!\Gamma\left(\varepsilon\right)}\frac{\Gamma\left(m+\varepsilon\right)}{m!\Gamma\left(\varepsilon\right)}}\int_{0}^{1}\int_{0}^{2\pi}\left(1-r_{\zeta}^{2}\right)^{\varepsilon}\left(-r_{\zeta}\right)^{n}\left(-r_{\zeta}\right)^{m}\times$
$\displaystyle
w\left(\zeta\right)\exp\left[i\left(n-m\right)\theta_{\zeta}\right]r_{\zeta}dr_{\zeta}d\theta_{\zeta}=1.$
(32)

On the other hand, we may readily show that
$w\left(\zeta\right)=w\left(r_{\zeta}\right)$, which implies that the integral
on $\theta_{\zeta}$ becomes

$\int_{0}^{2\pi}d\theta_{\zeta}\exp\left[i\left(n-m\right)\theta_{\zeta}\right]=2\pi\delta_{n,m}.$
(33)

From here, we can write (32) in the form

$2\pi{\displaystyle\sum\limits_{n=0}^{\infty}}\left|2n,\varepsilon\right\rangle\left\langle\varepsilon,2n\right|\frac{\Gamma\left(n+\varepsilon\right)}{n!\Gamma\left(\varepsilon\right)}\int_{0}^{1}\left(1-r_{\zeta}^{2}\right)^{\varepsilon}r_{\zeta}^{2n+1}w\left(r_{\zeta}\right)dr_{\zeta}=1.$
(34)

By using the following relationship among gamma functions

$\frac{\Gamma\left(x\right)\Gamma\left(y\right)}{\Gamma\left(x+y\right)}=2\int_{0}^{1}\left(1-t^{2}\right)^{y-1}t^{2x-1}dt,\text{
\ }\operatorname{Re}\left(x\right)>0,\text{ \
}\operatorname{Re}\left(y\right)>0,$ (35)

one can see that the weight function $w\left(r_{\zeta}\right)$ must have the
form

$w\left(r_{\zeta}\right)=\frac{\Gamma\left(\varepsilon\right)}{\pi\Gamma\left(\varepsilon-1\right)\left(1-r_{\zeta}^{2}\right)^{2}}=\frac{\varepsilon-1}{\pi\left(1-r_{\zeta}^{2}\right)^{2}},\text{
\ }\varepsilon>1,$ (36)

in order to ensure that the completeness relation (29) is satisfied. It is
important to highlight that the condition $\varepsilon>1$ leads to a positive
weight function $w\left(r_{\zeta}\right)$. At the same time, this condition
shows that the canonical algebra ($\varepsilon=1/2$) does not allow to obtain
a completeness relation for the generalized SVS. It follows from (27) and (29)
that these states form an overcomplete set of states on the Hilbert space. The
plot of weight function can be seen in the Fig. 2.

Figure 2: Graphics of $w\left(r_{\zeta}\right)$.

##  IV Time-dependent generalized CS

In Ref. (PER2021, ), the SCS has been built through the nonunitary approach by
considering the canonical algebra. In another way, by considering the WHA, it
is not possible to obtain the SCS since, from Baker–Campbell–Hausdorff
theorem, we cannot establish a direct relationship between the para-Bose
operators and integral of motion, due the deformed commutation relation.
However, instead of following the nonunitary approach, we will construct the
eigenstates of the integral of motion $\hat{A}$, in the form

$\hat{A}\left|z,t\right\rangle=z\left|z,t\right\rangle,$ (37)

where $z$ is a complex constant. The Eq. (37) allows us to assume

$\varphi_{0}=0\Longrightarrow u=0,$ (38)

without loss of generality.

The states $\left|z,t\right\rangle$ can be expanded in terms of the para-Bose
number states, as follows

$\left|z,t\right\rangle={\displaystyle\sum\limits_{n=0}^{\infty}}c_{n}\left|n,\varepsilon\right\rangle={\displaystyle\sum\limits_{n=0}^{\infty}}\left(c_{2n}\left|2n,\varepsilon\right\rangle+c_{2n+1}\left|2n+1,\varepsilon\right\rangle\right),$
(39)

where $c_{n}=c_{n}\left(t\right)$ are time-dependent coefficients and will be
determined such that the Eqs. (37) and (5) will be satisfied. Substituting
(6), (13) and (39) into (37), we find the following equations for the
coefficients

$\displaystyle
c_{2n}=\sqrt{\frac{n!\Gamma\left(\varepsilon\right)}{\Gamma\left(n+\varepsilon\right)}}\left(-\frac{g}{f}\right)^{n}L_{n}^{\varepsilon-1}\left(\frac{z^{2}}{2gf}\right)c_{0},\text{
\ }$ $\displaystyle
c_{2n+1}=\frac{z}{f}\sqrt{\frac{n!\Gamma\left(\varepsilon\right)}{2\Gamma\left(n+\varepsilon+1\right)}}\left(-\frac{g}{f}\right)^{n}L_{n}^{\varepsilon}\left(\frac{z^{2}}{2gf}\right)c_{0},$
(40)

where $L_{n}^{\alpha}\left(x\right)$ are the associated Laguerre polynomials
and $c_{0}$ is a time-dependent function, which will be determined such that
the states $\left|z,t\right\rangle$ satisfy the Schrödinger’s equation (5).

Here, it is convenient to introduce the squeeze $\zeta=g/f$ and displacement
$\xi=z/f$ parameters to rewrite the states
$\left|z,t\right\rangle\longrightarrow\left|\zeta,\xi\right\rangle$, in the
form

$\left|\zeta,\xi\right\rangle=\sqrt{\Gamma\left(\varepsilon\right)}c_{0}{\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta\right)^{n}\sqrt{n!}\left[\frac{L_{n}^{\varepsilon-1}\left(\frac{\xi^{2}}{2\zeta}\right)}{\sqrt{\Gamma\left(n+\varepsilon\right)}}\left|2n,\varepsilon\right\rangle+\frac{\xi
L_{n}^{\varepsilon}\left(\frac{\xi^{2}}{2\zeta}\right)}{\sqrt{2\Gamma\left(n+\varepsilon+1\right)}}\left|2n+1,\varepsilon\right\rangle\right],$
(41)

with $\zeta$ and $\xi$ satisfy the following differential equations:

$\dot{\zeta}=i\alpha^{\ast}\zeta^{2}-2i\beta\zeta+i\alpha,\text{ \
}\dot{\xi}=i\left(\alpha^{\ast}\zeta-\beta\right)\xi.$ (42)

From normalization condition

$\left\langle\xi,\zeta|\zeta,\xi\right\rangle=1,$ (43)

we found the following form for the function $c_{0}$,

$c_{0}=\left(\frac{\xi}{\sqrt{2}}\right)^{\varepsilon-1}\sqrt{\frac{1-\left|\zeta\right|^{2}}{\Gamma\left(\varepsilon\right)}}\frac{\exp\left\\{\frac{\zeta^{\ast}\xi^{2}}{2\left(1-\left|\zeta\right|^{2}\right)}+i\int\left[\operatorname{Re}\left(\alpha\zeta^{\ast}\right)-\beta\right]dt+i\phi\right\\}}{\sqrt{I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}},$
(44)

where $I_{\kappa}\left(Z\right)$ is the modified Bessel function of the first
kind, and $\phi$ is a time-dependent real function. Thus, the normalized
states $\left|\zeta,\xi\right\rangle$ take the form

$\displaystyle\left|\zeta,\xi\right\rangle$
$\displaystyle=\left(\frac{\xi}{\sqrt{2}}\right)^{\varepsilon-1}\sqrt{\frac{1-\left|\zeta\right|^{2}}{I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}}\exp\left\\{\frac{\zeta^{\ast}\xi^{2}}{2\left(1-\left|\zeta\right|^{2}\right)}+i\int\left[\operatorname{Re}\left(\alpha\zeta^{\ast}\right)-\beta\right]dt+i\phi\right\\}$
$\displaystyle\times{\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta\right)^{n}\sqrt{n!}\left[\frac{L_{n}^{\varepsilon-1}\left(\frac{\xi^{2}}{2\zeta}\right)}{\sqrt{\Gamma\left(n+\varepsilon\right)}}\left|2n,\varepsilon\right\rangle+\frac{\xi
L_{n}^{\varepsilon}\left(\frac{\xi^{2}}{2\zeta}\right)}{\sqrt{2\Gamma\left(n+\varepsilon+1\right)}}\left|2n+1,\varepsilon\right\rangle\right],$
(45)

Substituting $\left|\zeta,\xi\right\rangle$ into Schrödinger’s equation, we
find the following expression for $\phi$,

$\phi=-\int\delta dt.$ (46)

Therefore, the normalized states $\left|\zeta,\xi\right\rangle$ that satisfy
the Schrödinger’s equation are given by

$\displaystyle\left|\zeta,\xi\right\rangle$
$\displaystyle=\left(\frac{\xi}{\sqrt{2}}\right)^{\varepsilon-1}\sqrt{\frac{1-\left|\zeta\right|^{2}}{I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}}\exp\left[\frac{\zeta^{\ast}\xi^{2}}{2\left(1-\left|\zeta\right|^{2}\right)}+i\tilde{\vartheta}\right]$
$\displaystyle\times{\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta\right)^{n}\sqrt{n!}\left[\frac{L_{n}^{\varepsilon-1}\left(\frac{\xi^{2}}{2\zeta}\right)}{\sqrt{\Gamma\left(n+\varepsilon\right)}}\left|2n,\varepsilon\right\rangle+\frac{\xi
L_{n}^{\varepsilon}\left(\frac{\xi^{2}}{2\zeta}\right)}{\sqrt{2\Gamma\left(n+\varepsilon+1\right)}}\left|2n+1,\varepsilon\right\rangle\right],$
(47)

where

$\tilde{\vartheta}=\int\left[\operatorname{Re}\left(\alpha\zeta^{\ast}\right)-\beta-\delta\right]dt.$
(48)

In the following, we call the time-dependent states (47) generalized para-Bose
CS.

It’s worth mentioning that the states (47) present a set of information that,
to our knowledge, has not been previously derived, see, e.g., SHA1978 ;
SHA1981 ; BIS1980 ; SAX1986 ; MOJ2018a ; MOJ2018b ; Ald2017 ; Bag1997 ;
DEH2015 ; ALD2021 ; ALD2017 . Such information consists of the explicit form
of the squeeze, displacement, and deformation parameters (Wigner-parameter).
These parameters are present explicitly in the standard deviation and
probability transition. Furthermore, the time-dependent states (47) have been
expanded in terms of the time-independent para-Bose number states. Thus, the
states (47) lead to a deeper understanding of the physical systems described
by the Hamiltonian (4). As we will see from section V, the Hamiltonian can be
seen as a generalization of Calogero model CAL1971 .

In particular, taking into account the condition
$\zeta=0\Longrightarrow\alpha=0$, the states (47) take the form

$\displaystyle\left|\xi\right\rangle$
$\displaystyle=\left(\frac{\xi}{\sqrt{2}}\right)^{\varepsilon-1}\frac{\exp\left[-i\int\left(\beta+\delta\right)dt\right]}{\sqrt{I_{\varepsilon-1}\left(\left|\xi\right|^{2}\right)+I_{\varepsilon}\left(\left|\xi\right|^{2}\right)}}{\displaystyle\sum\limits_{n=0}^{\infty}}\left(\frac{\xi^{2}}{2}\right)^{n}\left[\frac{\left|2n,\varepsilon\right\rangle}{\sqrt{n!\Gamma\left(n+\varepsilon\right)}}+\frac{\xi\left|2n+1,\varepsilon\right\rangle}{\sqrt{2n!\Gamma\left(n+\varepsilon+1\right)}}\right]$
$\displaystyle=\sqrt{\Gamma\left(\varepsilon\right)}\exp\left[-i\int\left(\beta+\delta\right)dt\right]\left(\frac{\hat{a}^{\dagger}}{\sqrt{2}}\right)^{1-\varepsilon}\frac{I_{\varepsilon-1}\left(\xi\hat{a}^{\dagger}\right)+I_{\varepsilon}\left(\xi\hat{a}^{\dagger}\right)}{\sqrt{I_{\varepsilon-1}\left(\left|\xi\right|^{2}\right)+I_{\varepsilon}\left(\left|\xi\right|^{2}\right)}}\left|0,\varepsilon\right\rangle,$
(49)

with $\dot{\xi}=-i\beta\xi$ and
$\left|\xi\right\rangle=\left|0,\xi\right\rangle$. Except for time evolution,
these states correspond to the para-Bose CS obtained in Ref. SHA1978 . In
turn, we have that the condition $\varepsilon=1/2$ reduces the states (49) to
the form:

$\left|\xi\right\rangle=\exp\left(\int\frac{\beta+2\delta}{2i}dt-\frac{\left|\xi\right|^{2}}{2}\right)\exp\left(\xi\hat{a}^{\dagger}\right)\left|0\right\rangle,\text{
\ }\left|0\right\rangle=\left|0,\varepsilon=1/2\right\rangle,$ (50)

which are the time-dependent canonical CS GLA1966 ; MEH1966 .

The overlap of the states $\left|\zeta,\xi\right\rangle$ for different squeeze
$\zeta$ and displacement $\xi$ parameters is given by

$\displaystyle\left\langle\xi_{1},\zeta_{1}|\zeta,\xi\right\rangle$
$\displaystyle=\left(\frac{\xi\xi_{1}^{\ast}}{2}\right)^{\varepsilon-1}\sqrt{\frac{\left(1-\left|\zeta\right|^{2}\right)\left(1-\left|\zeta_{1}\right|^{2}\right)}{\left[I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)\right]\left[I_{\varepsilon-1}\left(\frac{\left|\xi_{1}\right|^{2}}{1-\left|\zeta_{1}\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi_{1}\right|^{2}}{1-\left|\zeta_{1}\right|^{2}}\right)\right]}}$
$\displaystyle\times\exp\left[\frac{\zeta^{\ast}\xi^{2}\left(1-\left|\zeta_{1}\right|^{2}\right)+\zeta_{1}\xi_{1}^{\ast
2}\left(1-\left|\zeta\right|^{2}\right)}{2\left(1-\left|\zeta\right|^{2}\right)\left(1-\left|\zeta_{1}\right|^{2}\right)}+i\int\operatorname{Re}\left(\alpha\zeta^{\ast}-\alpha\zeta_{1}^{\ast}\right)dt\right]$
$\displaystyle\times{\displaystyle\sum\limits_{n=0}^{\infty}}\left(\zeta\zeta_{1}^{\ast}\right)^{n}n!\left[\frac{L_{n}^{\varepsilon-1}\left(\frac{\xi^{2}}{2\zeta}\right)L_{n}^{\varepsilon-1}\left(\frac{\xi_{1}^{\ast
2}}{2\zeta_{1}^{\ast}}\right)}{\Gamma\left(n+\varepsilon\right)}+\frac{\xi\xi_{1}^{\ast}}{2}\frac{L_{n}^{\varepsilon}\left(\frac{\xi^{2}}{2\zeta}\right)L_{n}^{\varepsilon}\left(\frac{\xi_{1}^{\ast
2}}{2\zeta_{1}^{\ast}}\right)}{\Gamma\left(n+\varepsilon+1\right)}\right].$
(51)

Here, we can analyze the probability transition
$P_{n}\left(\zeta,\xi,\varepsilon\right)=\left|\left\langle\varepsilon,n|\zeta,\xi\right\rangle\right|^{2}$
of the number states $\left|n,\varepsilon\right\rangle$ to the CS
$\left|\zeta,\xi\right\rangle$:

$\displaystyle
P_{n}\left(\zeta,\xi,\varepsilon\right)=\left(\frac{\left|\xi\right|^{2}}{2}\right)^{\varepsilon-1}\frac{\left(1-\left|\zeta\right|^{2}\right)\exp\left[\frac{\operatorname{Re}\left(\zeta^{\ast}\xi^{2}\right)}{1-\left|\zeta\right|^{2}}\right]}{I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}m!\left|\zeta\right|^{2m}\left[\frac{\left|L_{m}^{\varepsilon-1}\left(\frac{\xi^{2}}{2\zeta}\right)\right|^{2}\delta_{n,2m}}{\Gamma\left(m+\varepsilon\right)}+\right.$
$\displaystyle\left.+\frac{\left|\xi\right|^{2}\left|L_{m}^{\varepsilon}\left(\frac{\xi^{2}}{2\zeta}\right)\right|^{2}\delta_{n,2m+1}}{2\Gamma\left(m+\varepsilon+1\right)}\right]$
(52)

The probability transition $P_{n}\left(\zeta,\xi,\varepsilon\right)$ has been
shown in Fig. 3. As we can see, the $\xi$-parameter allows access to odd
states while the $\varepsilon$ controls the dispersion of the probability
density. Furthermore, higher values of the $\varepsilon$-parameter allow
access to states with higher principal quantum number $n$, and attenuate the
access to the odd states.

(a)  (b)  (c)  (d)

Figure 3: Probability transition of the generalized CS.

###  IV.1 Mean values and uncertainty relations

Let us now consider the mean value of the momentum $\hat{P}$ and position
$\hat{x}$ operators, which satisfy the commutation relation

$\left[\hat{x},\hat{P}\right]=i\hbar\left[1+\left(2\varepsilon-1\right)\hat{R}\right],$
(53)

in the generalized CS (47). For this, we will rewrite these operators in terms
of the integrals of motion, as seen below

$\displaystyle\hat{a}=\frac{\hat{A}_{f}-\zeta\hat{A}_{f^{\ast}}^{\dagger}}{1-\left|\zeta\right|^{2}},\text{
\ }\hat{A}_{f}\equiv\frac{1}{f}\hat{A},$
$\displaystyle\hat{x}=\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}}l=\frac{l}{\sqrt{2}}\frac{\left(1-\zeta^{\ast}\right)\hat{A}_{f}+\left(1-\zeta\right)\hat{A}_{f^{\ast}}^{\dagger}}{1-\left|\zeta\right|^{2}},\text{
}$ (54)
$\displaystyle\hat{P}=\hbar\frac{\hat{a}-\hat{a}^{\dagger}}{i\sqrt{2}l}=\frac{\hbar}{i\sqrt{2}l}\frac{\left(1+\zeta^{\ast}\right)\hat{A}_{f}-\left(1+\zeta\right)\hat{A}_{f^{\ast}}^{\dagger}}{1-\left|\zeta\right|^{2}},$

where $l$ is a length-dimensional parameter which is related to the initial
standard deviation PER2021 . The new operator $\hat{A}_{f}$ acts on states
$\left|\zeta,\xi\right\rangle$ as follows

$\hat{A}_{f}\left|\zeta,\xi\right\rangle=\xi\left|\zeta,\xi\right\rangle.$
(55)

Using the relations (54) and (55), we can easily calculate the mean values of
the operators $\hat{x}$ and $\hat{P}$,

$\displaystyle\overline{x}$
$\displaystyle=\overline{x}\left(t\right)=\left\langle\xi,\zeta\left|\hat{x}\right|\zeta,\xi\right\rangle=\sqrt{2}l\frac{\operatorname{Re}\left[\left(1-\zeta^{\ast}\right)\xi\right]}{1-\left|\zeta\right|^{2}},$
$\displaystyle\overline{P}$
$\displaystyle=\overline{P}\left(t\right)=\left\langle\xi,\zeta\left|\hat{P}\right|\zeta,\xi\right\rangle=\frac{\sqrt{2}\hbar}{l}\frac{\operatorname{Im}\left[\left(1+\zeta^{\ast}\right)\xi\right]}{1-\left|\zeta\right|^{2}}.$
(56)

From here, there is a correspondence between the squeeze $\zeta$ and
displacement $\xi$ parameters with the mean values of $\overline{x}$ and
$\overline{P}$,

$\xi=\frac{1+\zeta}{\sqrt{2}l}\overline{x}+\frac{il}{\hbar}\frac{1-\zeta}{\sqrt{2}}\overline{P}.$
(57)

Taking the square of the operators $\hat{x}$ and $\hat{P}$, we have

$\displaystyle\hat{x}^{2}=\frac{l^{2}}{2}\frac{\left(1-\zeta^{\ast}\right)^{2}\hat{A}_{f}^{2}+\left(1-\zeta\right)^{2}\hat{A}_{f^{\ast}}^{\dagger
2}+2\left|1-\zeta\right|^{2}\hat{A}_{f^{\ast}}^{\dagger}\hat{A}_{f}}{\left(1-\left|\zeta\right|^{2}\right)^{2}}+l^{2}\frac{\left|1-\zeta\right|^{2}}{1-\left|\zeta\right|^{2}}\frac{1+\left(2\varepsilon-1\right)\hat{R}}{2},$
$\displaystyle\hat{P}^{2}=-\frac{\hbar^{2}}{2l^{2}}\frac{\left(1+\zeta^{\ast}\right)^{2}\hat{A}_{f}^{2}+\left(1+\zeta\right)^{2}\hat{A}_{f^{\ast}}^{\dagger
2}-2\left|1+\zeta\right|^{2}\hat{A}_{f^{\ast}}^{\dagger}\hat{A}_{f}}{\left(1-\left|\zeta\right|^{2}\right)^{2}}+\frac{\hbar^{2}}{l^{2}}\frac{\left|1+\zeta\right|^{2}}{1-\left|\zeta\right|^{2}}\frac{1+\left(2\varepsilon-1\right)\hat{R}}{2}.$
(58)

So, from these results, we can calculate the mean values of $\overline{x^{2}}$
and $\overline{P^{2}}$, as follow

$\displaystyle\overline{x^{2}}$
$\displaystyle=\overline{x^{2}}\left(t\right)=\left\langle\xi,\zeta\left|\hat{x}^{2}\right|\zeta,\xi\right\rangle=\frac{2l^{2}\operatorname{Re}^{2}\left[\left(1-\zeta^{\ast}\right)\xi\right]}{\left(1-\left|\zeta\right|^{2}\right)^{2}}+l^{2}\frac{\left|1-\zeta\right|^{2}}{1-\left|\zeta\right|^{2}}\frac{1+\left(2\varepsilon-1\right)\overline{R}}{2},$
$\displaystyle\overline{P^{2}}$
$\displaystyle=\overline{P^{2}}\left(t\right)=\left\langle\xi,\zeta\left|\hat{P}^{2}\right|\zeta,\xi\right\rangle=\frac{2\hbar^{2}\operatorname{Im}^{2}\left[\left(1+\zeta^{\ast}\right)\xi\right]}{l^{2}\left(1-\left|\zeta\right|^{2}\right)^{2}}+\frac{\hbar^{2}}{l^{2}}\frac{\left|1+\zeta\right|^{2}}{1-\left|\zeta\right|^{2}}\frac{1+\left(2\varepsilon-1\right)\overline{R}}{2},$
(59)

where

$\overline{R}=\left\langle\xi,\zeta\left|\hat{R}\right|\zeta,\xi\right\rangle=\frac{I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)-I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}{I_{\varepsilon-1}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{\varepsilon}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}.$
(60)

Note that $\xi=0$ leads to a well-defined even parity of the states (47), once
$\overline{R}=1$.

In what follows, we calculate the standard deviation, i.e.,

$\displaystyle\sigma_{x}$
$\displaystyle=\sigma_{x}\left(t\right)=\sqrt{\overline{x^{2}}-\overline{x}^{2}}=l\left|1-\zeta\right|\sqrt{\frac{1+\left(2\varepsilon-1\right)\overline{R}}{2\left(1-\left|\zeta\right|^{2}\right)}},$
$\displaystyle\sigma_{P}$
$\displaystyle=\sigma_{P}\left(t\right)=\sqrt{\overline{P^{2}}-\overline{P}^{2}}=\frac{\hbar}{l}\left|1+\zeta\right|\sqrt{\frac{1+\left(2\varepsilon-1\right)\overline{R}}{2\left(1-\left|\zeta\right|^{2}\right)}}.$
(61)

From Eqs. (61), it is easy to see that standard deviations in position and
momentum present the squeezing property.

Finally, we aim to find the Heisenberg uncertainty relation in the generalized
states (47) by considering the WHA. Therefore, in the hold of previous
results, we can explicitly calculate the product $\sigma_{x}\sigma_{P}$, as
shown below:

$\sigma_{x}\sigma_{P}=\hbar\frac{\left|1-\zeta\right|\left|1+\zeta\right|}{1-\left|\zeta\right|^{2}}\frac{1+\left(2\varepsilon-1\right)\overline{R}}{2}=\hbar\sqrt{1+\frac{4\operatorname{Im}^{2}\left(\zeta\right)}{\left(1-\left|\zeta\right|^{2}\right)^{2}}}\frac{1+\left(2\varepsilon-1\right)\overline{R}}{2}.$
(62)

In particular, if we consider that the squeeze parameter assumes real values,
this implies that
$\sigma_{x}\sigma_{P}=\hbar/2\left(1+\left(2\varepsilon-1\right)\overline{R}\right)$.
In turn, for this particular choice, we should note that the Heisenberg
uncertainty assumes the minimum value predicted by the relation

$\sigma_{A}\sigma_{B}\geq\frac{1}{2}\left|\left\langle\left[\hat{A},\hat{B}\right]\right\rangle\right|.$
(63)

It must be highlighted that in Ref. (SHA1981, ), a detailed analysis has been
applied to show how the para-Bose uncertainty relation leads to uncertainty
relation computed from canonical commutation relation.

Taking into account the covariance $\sigma_{xP}$,

$\sigma_{xP}=\frac{\left\langle\xi,\zeta\left|\hat{P}\hat{x}\right|\zeta,\xi\right\rangle+\left\langle\xi,\zeta\left|\hat{x}\hat{P}\right|\zeta,\xi\right\rangle}{2}-\overline{x}\left(t\right)\overline{P}\left(t\right)=-\hbar\operatorname{Im}\left(\zeta\right)\frac{1+\left(2\varepsilon-1\right)\overline{R}}{1-\left|\zeta\right|^{2}},$
(64)

we can calculate the Schrödinger-Robertson uncertainty relation (ROB1930, ),

$\sigma_{x}^{2}\sigma_{P}^{2}-\sigma_{xP}^{2}=\frac{\hbar^{2}}{4}\left[1+\left(2\varepsilon-1\right)\overline{R}\right]^{2},$
(65)

which, as can we see, is minimized.

##  V Coordinate representation of the generalized CS

It is well-known that the phase space in the context of quantum mechanics
encounters difficulties due to the uncertainty principle. Considering the
states $\left|\zeta,\xi\right\rangle$ in a coordinate representation allow us
to introduce the quasiprobability Wigner distribution, which plays an
analogous role to the classical distributions (HIL1984, ; WIL2007, ; OLI2012,
).

According to WHA, $\hat{P}$ is a self-adjoint operator on semi-axis
$\left(x\geq 0\right)$, see (MUK1980, ), and have the following form:

$\hat{P}=-i\hbar\partial_{x}+\frac{i\hbar}{2x}\left(2\varepsilon-1\right)\hat{R}.$
(66)

In this case, the annihilation operator takes the form

$\hat{a}=\frac{1}{\sqrt{2}}\left(\frac{\hat{x}}{l}+\frac{il}{\hbar}\hat{P}\right)=\frac{l}{\sqrt{2}}\left(\partial_{x}-\frac{2\varepsilon-1}{2x}\hat{R}+\frac{x}{l^{2}}\right).$
(67)

Applying the annihilation condition

$\hat{a}\Psi_{0,\varepsilon}\left(x\right)=0,\text{ \
}\Psi_{0,\varepsilon}\left(x\right)=\left\langle
x|0,\varepsilon\right\rangle,$ (68)

we can obtain a differential equation for the vacuum state, as follows:

$\left(\partial_{x}+\frac{x}{l^{2}}-\frac{2\varepsilon-1}{2x}\right)\Psi_{0,\varepsilon}\left(x\right)=0,\text{
\
}\hat{R}\Psi_{0,\varepsilon}\left(x\right)=\Psi_{0,\varepsilon}\left(x\right).$
(69)

The general solution reads

$\Psi_{0,\varepsilon}\left(x\right)=Cx^{\varepsilon-\frac{1}{2}}\exp\left(-\frac{x^{2}}{2l^{2}}\right),$
(70)

with $C$ being a real constant, which will be determined through the
normalization condition, as seen below,

$2C^{2}\int_{0}^{\infty}x^{2\varepsilon-1}\exp\left(-\frac{x^{2}}{l^{2}}\right)dx=1\Rightarrow
C=\frac{1}{l^{\varepsilon}\sqrt{\Gamma\left(\varepsilon\right)}}.$ (71)

It should be noted that the condition (69) leads to the following quantization
condition:

$\varepsilon=2\ell+\frac{1}{2},\text{ \ }\ell=0,1,2,\ldots,$ (72)

where $\ell$ is analogous to the angular momentum. Such quantization was
obtained in (LOH2004, ) to ensure that the eigenfunctions of $\hat{P}$ are
differentiable at the origin.

Thus, the vacuum state with even parity
$\Psi_{0,\varepsilon}\left(x\right)\equiv\Psi_{0,\ell}^{e}\left(x\right)$
takes the form

$\Psi_{0,\ell}^{e}\left(x\right)=\frac{1}{l^{2\ell+\frac{1}{2}}\sqrt{\Gamma\left(2\ell+\frac{1}{2}\right)}}x^{2\ell}\exp\left(-\frac{x^{2}}{2l^{2}}\right).$
(73)

Taking into account that $\Psi_{\zeta,\xi}^{\ell}\left(x,t\right)=\left\langle
x|\zeta,\xi\right\rangle$ and replacing the relations (14) in the states (47),
we obtain

$\displaystyle\Psi_{\zeta,\xi}^{\ell}\left(x,t\right)=\xi^{2\ell-\frac{1}{2}}\frac{\sqrt{\left(1-\left|\zeta\right|^{2}\right)\Gamma\left(2\ell+\frac{1}{2}\right)}\exp\left[\frac{\zeta^{\ast}\xi^{2}}{2\left(1-\left|\zeta\right|^{2}\right)}+i\tilde{\vartheta}\right]}{2^{\frac{4\ell-1}{4}}\sqrt{I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}}$
$\displaystyle\times{\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\frac{\zeta\hat{a}^{\dagger
2}}{2}\right)^{n}\left[\frac{L_{n}^{2\ell-\frac{1}{2}}\left(\frac{\xi^{2}}{2\zeta}\right)}{\Gamma\left(n+2\ell+\frac{1}{2}\right)}+\frac{L_{n}^{2\ell+\frac{1}{2}}\left(\frac{\xi^{2}}{2\zeta}\right)}{\Gamma\left(n+2\ell+\frac{3}{2}\right)}\frac{\xi\hat{a}^{\dagger}}{2}\right]\Psi_{0,\ell}^{e}\left(x\right).$
(74)

Using the results below

$\displaystyle\left(\hat{a}^{\dagger}\right)^{2n}\Psi_{0,\ell}^{e}\left(x\right)=\left(-1\right)^{n}2^{n}n!L_{n}^{2\ell-\frac{1}{2}}\left(\frac{x^{2}}{l^{2}}\right)\Psi_{0,\ell}^{e}\left(x\right),$
$\displaystyle\left(\hat{a}^{\dagger}\right)^{2n+1}\Psi_{0,\ell}^{e}\left(x\right)=\frac{\sqrt{2}x}{l}\left(-1\right)^{n}2^{n}n!L_{n}^{2\ell+\frac{1}{2}}\left(\frac{x^{2}}{l^{2}}\right)\Psi_{0,\ell}^{e}\left(x\right),$
(75)

we can write (74), as follows

$\displaystyle\Psi_{\zeta,\xi}^{\ell}\left(x,t\right)=\left\langle
x|\zeta,\xi\right\rangle=\frac{\sqrt{1-\left|\zeta\right|^{2}}}{1-\zeta}\frac{\sqrt{x}}{l}\frac{I_{2\ell-\frac{1}{2}}\left(\frac{\sqrt{2}\xi}{1-\zeta}\frac{x}{l}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\sqrt{2}\xi}{1-\zeta}\frac{x}{l}\right)}{\sqrt{I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)}}$
$\displaystyle\times\exp\left[-\frac{1+\zeta}{1-\zeta}\frac{x^{2}}{2l^{2}}-\frac{\left(1-\zeta^{\ast}\right)\xi^{2}}{2\left(1-\zeta\right)\left(1-\left|\zeta\right|^{2}\right)}+i\tilde{\vartheta}\right].$
(76)

In particular, if $\ell=0$ we obtain the following result:

$\displaystyle\Psi_{\zeta,\xi}^{0}\left(x,t\right)=\frac{\left(1-\left|\zeta\right|^{2}\right)^{1/4}}{\sqrt{\sqrt{\pi}l\left(1-\zeta\right)}}\exp\left[-\frac{1}{2l^{2}}\frac{1+\zeta}{1-\zeta}\left(x-\frac{l\sqrt{2}\xi}{1+\zeta}\right)^{2}+\frac{\left(1+\zeta^{\ast}\right)}{\left(1+\zeta\right)\left(1-\left|\zeta\right|^{2}\right)}\frac{\xi^{2}}{2}-\frac{1}{2}\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}+i\varrho\right],$
$\displaystyle\varrho=\frac{1}{2}\int\left[\operatorname{Re}\left(\alpha\zeta^{\ast}-\beta\right)-2\delta\right]dt.$
(77)

Performing the following identifications $\zeta=g/f$, $\xi=z/f=-\varphi/f$ and
$\mu=\left|f\right|^{2}\left(1-\left|\zeta\right|^{2}\right)=1$ leads to the
results of the recent publication (PER2021, ).

The probability density that corresponds to the generalized CS is given by

$\displaystyle\rho_{\zeta,\xi}^{\ell}\left(x,t\right)=\left|\Psi_{\zeta,\xi}^{\ell}\left(x,t\right)\right|^{2}=\frac{1-\left|\zeta\right|^{2}}{\left|1-\zeta\right|^{2}}\frac{x}{l^{2}}\frac{\left|I_{2\ell-\frac{1}{2}}\left(\frac{\sqrt{2}\xi}{1-\zeta}\frac{x}{l}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\sqrt{2}\xi}{1-\zeta}\frac{x}{l}\right)\right|^{2}}{\left|I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi\right|^{2}}{1-\left|\zeta\right|^{2}}\right)\right|}$
$\displaystyle\times\exp\left[-\frac{1-\left|\zeta\right|^{2}}{\left|1-\zeta\right|^{2}}\frac{x^{2}}{l^{2}}-\frac{1}{1-\left|\zeta\right|^{2}}\operatorname{Re}\left(\frac{1-\zeta^{\ast}}{1-\zeta}\xi^{2}\right)\right].$
(78)

In Fig. 4, we have obtained some plots of the probability density. As we can
see, the probability density of the generalized CS has a shape of a Gaussian
distribution, which moves in space as $\ell$ increases.

Figure 4: Displayed is the probability density by assuming fixed values $l=1$,
$\zeta=0.45$ and $\xi=i$. For $\ell=0$, we have the SCS, while for $\ell>0$,
we obtained the generalized para-Bose CS.

Finally, substituting (54) into (4), the Hamiltonian takes the form

$\displaystyle\hat{H}$
$\displaystyle=\frac{1}{2}\hbar\left(\alpha^{\ast}\hat{a}^{2}+\alpha\hat{a}^{\dagger
2}\right)+\frac{1}{2}\hbar\beta\left(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}\right)+\hbar\delta$
$\displaystyle=\frac{\hat{P}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}^{2}+\frac{1}{2}\Omega\left(\hat{P}\hat{x}+\hat{x}\hat{P}\right)+\mathcal{E},$
$\displaystyle=-\frac{\hbar^{2}}{2m}\partial_{x}^{2}-i\hbar\Omega
x\partial_{x}+\frac{\hbar^{2}}{2mx^{2}}2\ell\left(2\ell-\hat{R}\right)+\frac{1}{2}m\omega^{2}x^{2}+\mathcal{E}-\frac{i\hbar\Omega}{2},$
(79)

where the time-dependent quantities $m$, $\omega$, $\Omega$ and $\mathcal{E}$
reads

$\displaystyle\frac{1}{m}$
$\displaystyle=\frac{l^{2}}{\hbar}\operatorname{Re}\left(\beta-\alpha\right),\text{
\
}m\omega^{2}=\frac{\hbar}{l^{2}}\operatorname{Re}\left(\beta+\alpha\right),\text{
\ }\Omega=\operatorname{Im}\left(\alpha\right),\text{ \
}\mathcal{E}=\hbar\delta,$ $\displaystyle\omega^{2}$
$\displaystyle=\beta^{2}-\operatorname{Re}^{2}\left(\alpha\right).$ (80)

From Hamiltonian (79), we can describe the physical systems such as harmonic
oscillators, which are described by confining potential $V_{HO}\sim x^{2}$. We
also can establish an analogy of the potential
$V_{C}=\frac{\hbar^{2}}{2mx^{2}}2\ell\left(2\ell-\hat{R}\right)$ with the
centrifugal potential. On the other hand, one may identify the potential
$V_{C}$ with the conformal sector of quantum conformal mechanics, except by
the $\hat{R}$ operator AND2016 . Since the reflection operator acting on even
parity states leads to $\hat{R}\psi_{e}=\psi_{e}$, it must be noticed that the
potential $V_{C}$ has a negative correction term owing to the reflection
operator. Furthermore, the repulsiveness of the $V_{C}$ potential is weakened
by the correction arising from the reflection operator.

Finally, we can relate the action of the reflection operator on a particular
state to the reduction of the dimensionality of space. Let us analyze the
centrifugal term of the $d$-dimensional Laplacian, which can be written in the
form LOH2004 :

$\Delta^{\left(d\right)}=\partial_{r}^{2}+\frac{d-1}{r}\partial_{r}+\frac{L\left(L+d-2\right)}{r^{2}}.$
(81)

At this point, we can establish a direct relationship between the action of
the reflection operator on even parity states and the reduction of the space
from three to one dimension $\left(d=1\right)$, from the perspective of the
centrifugal term by considering that $L=2\ell$.

##  VI Time-independent para-Bose Oscillator

Since the results obtained in this work can be applied to both time-dependent
and time-independent physical systems, for the sake of simplicity, we apply
this approach to the study of the time-independent para-Bose oscillator.
First, we obtain the Hamiltonian of the time-independent para-Bose oscillator
applying the following conditions $\alpha=\delta=\Omega=0$ and
$\beta=\beta_{0}$ on the Eq. (4), which can be written in the form:

$\displaystyle\hat{H}=\frac{1}{2}\hbar\beta_{0}\left(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}\right)=\frac{\hbar^{2}}{2m_{0}}\left[-\partial_{x}^{2}+\frac{2\ell\left(2\ell-\hat{R}\right)}{x^{2}}+\frac{m_{0}^{2}\omega_{0}^{2}}{\hbar^{2}}x^{2}\right],$
$\displaystyle\omega_{0}=\beta_{0},\text{ \
}m_{0}=\frac{\hbar}{l^{2}\beta_{0}},$ (82)

where the subindex labels the initial time, and from (72) we can write the
para-Bose number states
$\left|n,\varepsilon\right\rangle\Longrightarrow\left|n,\ell\right\rangle$,
such that

$\hat{H}\left|n,\ell\right\rangle=\hbar\omega_{0}\left(n+2\ell+\frac{1}{2}\right)\left|n,\ell\right\rangle.$
(83)

Notice that, taking $\ell=0$ lead to the standard harmonic oscillator.

In this case, the equation system (8), take the form

$\dot{f}=i\omega_{0}f,\text{ \ }\dot{g}=-i\omega_{0}g,\text{ \
}\dot{\varphi}=0,$ (84)

whose solution is given by

$f=f_{0}e^{i\omega_{0}t},\text{ \ }g=g_{0}e^{-i\omega_{0}t},\text{ \
}\varphi\equiv 0.$ (85)

From here, we can write the squeeze and displacement parameters in the form

$\displaystyle\zeta=\frac{g}{f}=\zeta_{0}e^{-2i\omega_{0}t},\text{ \
}\xi=\frac{z}{f}=\xi_{0}e^{-i\omega_{0}t},$
$\displaystyle\zeta_{0}=\left|\zeta_{0}\right|e^{i\theta_{\zeta}}=\frac{g_{0}}{f_{0}},\text{
\ }\xi_{0}=\left|\xi_{0}\right|e^{i\theta_{\xi}}=\frac{z}{f_{0}}.$ (86)

In the following, we can rewrite the operators (54):

$\displaystyle\hat{a}=\frac{\hat{A}_{f}-\zeta_{0}e^{-2i\omega_{0}t}\hat{A}_{f^{\ast}}^{\dagger}}{1-\left|\zeta_{0}\right|^{2}},\text{
\ }\hat{A}_{f}\equiv\frac{1}{f_{0}e^{i\omega_{0}t}}\hat{A},$
$\displaystyle\hat{x}=\frac{l}{\sqrt{2}}\frac{\left(1-\zeta_{0}^{\ast}e^{2i\omega_{0}t}\right)\hat{A}_{f}+\left(1-\zeta_{0}e^{-2i\omega_{0}t}\right)\hat{A}_{f^{\ast}}^{\dagger}}{1-\left|\zeta_{0}\right|^{2}},\text{
}$
$\displaystyle\hat{P}=\frac{\hbar}{i\sqrt{2}l}\frac{\left(1+\zeta_{0}^{\ast}e^{2i\omega_{0}t}\right)\hat{A}_{f}-\left(1+\zeta_{0}e^{-2i\omega_{0}t}\right)\hat{A}_{f^{\ast}}^{\dagger}}{1-\left|\zeta_{0}\right|^{2}},$
(87)

and the generalized CS (47), as follows

$\displaystyle\left|\zeta,\xi\right\rangle=\left(\frac{\left|\xi_{0}\right|e^{i\theta_{\xi}}}{\sqrt{2}}\right)^{2\ell-\frac{1}{2}}\sqrt{\frac{1-\left|\zeta_{0}\right|^{2}}{I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\left|\zeta_{0}\right|^{2}}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\left|\zeta_{0}\right|^{2}}\right)}}\exp\left[\frac{\left|\zeta_{0}\right|\left|\xi_{0}\right|^{2}e^{2i\theta_{\xi}-i\theta_{\zeta}}}{2\left(1-\left|\zeta_{0}\right|^{2}\right)}\right]\times$
$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}\left(-\zeta_{0}\right)^{n}\sqrt{n!}e^{-2i\omega_{0}\left(n+\ell\right)t}\left[\frac{L_{n}^{2\ell-\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}e^{2i\theta_{\xi}}}{2\left|\zeta_{0}\right|e^{i\theta_{\zeta}}}\right)e^{-\frac{i\omega_{0}t}{2}}}{\sqrt{\Gamma\left(n+2\ell+\frac{1}{2}\right)}}\left|2n,\ell\right\rangle+\frac{\left|\xi_{0}\right|e^{i\theta_{\xi}}L_{n}^{2\ell+\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}e^{2i\theta_{\xi}}}{2\left|\zeta_{0}\right|e^{i\theta_{\zeta}}}\right)e^{-\frac{3i\omega_{0}}{2}t}}{\sqrt{2\Gamma\left(n+2\ell+\frac{3}{2}\right)}}\left|2n+1,\ell\right\rangle\right].$
(88)

From Eq. (56), we get $\overline{x}$ and $\overline{P}$ in the form

$\displaystyle\overline{x}=\overline{x}_{0}\cos\left(\omega_{0}t\right)+\frac{\overline{P}_{0}}{m_{0}\omega_{0}}\sin\left(\omega_{0}t\right),\text{
\
}\overline{P}=\overline{P}_{0}\cos\left(\omega_{0}t\right)-m_{0}\omega_{0}\overline{x}_{0}\sin\left(\omega_{0}t\right),$
$\displaystyle\overline{x}_{0}=\sqrt{2}l\left|\xi_{0}\right|\frac{\cos\left(\theta_{\xi}\right)-\left|\zeta_{0}\right|\cos\left(\theta_{\xi}-\theta_{\zeta}\right)}{1-\left|\zeta_{0}\right|^{2}},\text{
\
}\overline{P}_{0}=\sqrt{2}lm_{0}\omega_{0}\left|\xi_{0}\right|\frac{\sin\left(\theta_{\xi}\right)+\left|\zeta_{0}\right|\sin\left(\theta_{\xi}-\theta_{\zeta}\right)}{1-\left|\zeta_{0}\right|^{2}}.$
(89)

Notice that taking $\xi_{0}=0$, the mean values of $\overline{x}$ and
$\overline{P}$ are equal to zero, which corresponds to the mean values
evaluated in the generalized SVS.

The uncertainty relation (62) and (65) becomes

$\displaystyle\sigma_{x}\sigma_{P}=\hbar\sqrt{1+\frac{4\left|\zeta_{0}\right|^{2}\sin^{2}\left(\theta_{\zeta}-2\omega_{0}t\right)}{\left(1-\left|\zeta_{0}\right|^{2}\right)^{2}}}\frac{1+4\ell\overline{R}}{2},$
$\displaystyle\sigma_{x}^{2}\sigma_{P}^{2}-\sigma_{xP}^{2}=\frac{\hbar^{2}}{4}\left(1+4\ell\overline{R}\right)^{2},$
(90)

with the mean value of $\hat{R}$ is given by the Eq. (60),

$\overline{R}=\frac{I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\left|\zeta_{0}\right|^{2}}\right)-I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\left|\zeta_{0}\right|^{2}}\right)}{I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\left|\zeta_{0}\right|^{2}}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\left|\zeta_{0}\right|^{2}}\right)}.$
(91)

One can see that the uncertainty relation has an oscillatory behavior,
reaching minimum values at specific points given by

$\sin\left(\theta_{\zeta}-2\omega_{0}t_{k}\right)=0,\text{ \
}t_{k}=\frac{\theta_{\zeta}-k\pi}{2\omega_{0}},$ (92)

where $k=0,\pm 1,\pm 2,\ldots$. Here $t_{k}$ corresponds to the values on the
time for which the uncertainty relation is minimal. The condition
$\theta_{\zeta}=0\Longrightarrow\zeta_{0}=\zeta_{0}^{\ast}$ leads to minimum
uncertainty at $t=0$. From Eqs. (61), (72) and (90), we can write the
$l$-parameter in term of the quantities
$\sigma_{x}\left(0\right)=\sigma_{x_{0}}$, $\zeta_{0}$, $\xi_{0}$ and $\ell$,
as follows

$l=\sqrt{\frac{1+\zeta_{0}}{1-\zeta_{0}}\frac{2}{1+4\ell\overline{R}}}\sigma_{x_{0}},\text{
\ }\sigma_{x_{0}}=\hbar\frac{1+4\ell\overline{R}}{2\sigma_{P_{0}}}.$ (93)

In what the following, let us consider $\zeta_{0}$ as being a real parameter.

It must be highlighted that the para-Bose number states are eigenstates of the
Hamiltonian (82), and therefore the standard deviation for this operator is
null when evaluated on this basis. Since $\hat{H}$ is time-independent, we
have that $\left|n,\ell\right\rangle$ are eigenstates of $\hat{H}$ with well-
defined energy. Furthermore, it is interesting to verify the probability
transition from the states with well-defined energy to generalized CS (88).
Therefore, it follows from (52), (72) and (86) that

$\displaystyle
P_{n}\left(\zeta_{0},\xi_{0},\ell\right)=\left(\frac{\left|\xi_{0}\right|^{2}}{2}\right)^{2\ell-\frac{1}{2}}\frac{\left(1-\zeta_{0}^{2}\right)\exp\left[\frac{\zeta_{0}\left|\xi_{0}\right|^{2}}{1-\zeta_{0}^{2}}\cos\left(2\theta_{\xi}\right)\right]}{I_{2\ell-\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\zeta_{0}^{2}}\right)+I_{2\ell+\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}}{1-\zeta_{0}^{2}}\right)}n!\zeta_{0}^{2n}$
$\displaystyle\times\left[\frac{\left|L_{n}^{2\ell-\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}e^{2i\theta_{\xi}}}{2\zeta_{0}}\right)\right|^{2}}{\Gamma\left(n+2\ell+\frac{1}{2}\right)}+\frac{\left|\xi_{0}\right|^{2}\left|L_{n}^{2\ell+\frac{1}{2}}\left(\frac{\left|\xi_{0}\right|^{2}e^{2i\theta_{\xi}}}{2\zeta_{0}}\right)\right|^{2}}{2\Gamma\left(n+2\ell+\frac{3}{2}\right)}\right].$
(94)

Notice that the probability transition is time-independent. As we saw in the
figures Fig. 1 and Fig. 3, the probability transition has its shape
significantly altered by the displacement and Wigner parameters. In Fig. 5, we
will make an analysis considering some values for the squeeze parameter
$\zeta_{0}$, keeping the other parameters fixed; namely, $\xi_{0}$ and $\ell$.

(a)  (b)  (c)  (d)

Figure 5: The probability transition is showed by considering fixed values
$\ell=2$, $\left|\xi_{0}\right|=1$, and $\theta_{\xi}=\pi/2$. These conditions
imply that $\overline{x}_{0}=0$ and
$\overline{P}_{0}=\sqrt{2}lm_{0}\omega_{0}\left|\xi_{0}\right|\left(1+\zeta_{0}\right)/\left(1-\zeta_{0}^{2}\right)$.
In Figure (a) we recover the para-Bose CS. Meanwhile, in Figures (b), (c) and
(d) we have the generalized para-Bose CS.

From Eqs. (89), we have that the $\zeta_{0}$,$\xi_{0}$-parameters are directly
related to the initial conditions that lead to the temporal evolution of the
mean values in the position and momentum. As we saw, the standard deviation
and the uncertainty relations are expressed in terms of special functions,
whose analysis is not straightforward. Therefore, it is interesting to
investigate the asymptotic form of Bessel functions of the first kind present
in uncertainty relations as follows:

a) Asymptotic form for small arguments $Z$ and fixed $\kappa$ (see Eq. 9.6.7,
page 375 in Ref. Abr1965 ).

$I_{\kappa}\left(Z\right)\sim\frac{1}{\Gamma\left(\kappa+1\right)}\left(\frac{Z}{2}\right)^{\kappa}.$
(95)

This limit implies that $\left|\xi_{0}\right|\rightarrow 0$ and $\zeta_{0}<1$.
In its turn, the mean value of the reflection operator $\overline{R}$ can be
rewritten:

$\overline{R}\sim\frac{\left(4\ell+1\right)\left(1-\zeta_{0}^{2}\right)-\left|\xi_{0}\right|^{2}}{\left(4\ell+1\right)\left(1-\zeta_{0}^{2}\right)+\left|\xi_{0}\right|^{2}},$
(96)

and the uncertainty relations (90) become

$\displaystyle\sigma_{x}\sigma_{P}\sim\frac{\hbar}{2}\sqrt{1+\frac{4\zeta_{0}^{2}\sin^{2}\left(2\omega_{0}t\right)}{\left(1-\zeta_{0}^{2}\right)^{2}}}\frac{\left(4\ell+1\right)^{2}\left(1-\zeta_{0}^{2}\right)-\left(4\ell-1\right)\left|\xi_{0}\right|^{2}}{\left(4\ell+1\right)\left(1-\zeta_{0}^{2}\right)+\left|\xi_{0}\right|^{2}},$
$\displaystyle\sigma_{x}^{2}\sigma_{P}^{2}-\sigma_{xP}^{2}\sim\frac{\hbar^{2}}{4}\left[\frac{\left(4\ell+1\right)^{2}\left(1-\zeta_{0}^{2}\right)-\left(4\ell-1\right)\left|\xi_{0}\right|^{2}}{\left(4\ell+1\right)\left(1-\zeta_{0}^{2}\right)+\left|\xi_{0}\right|^{2}}\right]^{2}.$
(97)

From (96), it is easy to see that the parity of the states is even when we
take $\left|\xi_{0}\right|=0$. On the other hand, the value of the
uncertainties, Eq. (97), increases as $\ell$ increases, while for
$\zeta_{0}=\ell=0$ is minimized.

b) Asymptotic form for large arguments $Z$ and fixed $\kappa$ (see Eq. 9.7.1,
page 377 in Ref. Abr1965 ),

$I_{\kappa}\left(Z\right)\sim\frac{e^{Z}}{\sqrt{2\pi
Z}}\left[1-\frac{1}{2Z}\left(\kappa^{2}-\frac{1}{4}\right)\right].$ (98)

This limit can be obtained in two different ways, first it is
$\left|\xi_{0}\right|\rightarrow\infty$ with $\zeta_{0}<1$ and second
$\zeta_{0}\rightarrow 1$ with $\left|\xi_{0}\right|<\infty$. In this second
case, we have a high degree of squeeze in $\sigma_{x}$, which becomes smaller
compared to $\sigma_{P}$, as $\zeta_{0}$ approaches $1$. From Eq. (98), we can
write the mean value of $\overline{R}$ as follows

$\overline{R}\sim\frac{\ell\left(1-\zeta_{0}^{2}\right)}{\left|\xi_{0}\right|^{2}-2\ell^{2}\left(1-\zeta_{0}^{2}\right)}.$
(99)

Note that the range of values
$\left|\xi_{0}\right|^{2}<2\ell^{2}\left(1-\zeta_{0}^{2}\right)$ takes
$\overline{R}<0$, indicating that only odd parity states stay at this limit.
The uncertainty relations can be rewritten in the form

$\displaystyle\sigma_{x}\sigma_{P}\sim\frac{\hbar}{2}\sqrt{1+\frac{4\zeta_{0}^{2}\sin^{2}\left(2\omega_{0}t\right)}{\left(1-\zeta_{0}^{2}\right)^{2}}}\frac{\left|\xi_{0}\right|^{2}+2\ell^{2}\left(1-\zeta_{0}^{2}\right)}{\left|\xi_{0}\right|^{2}-2\ell^{2}\left(1-\zeta_{0}^{2}\right)},$
$\displaystyle\sigma_{x}^{2}\sigma_{P}^{2}-\sigma_{xP}^{2}\sim\frac{\hbar^{2}}{4}\left[\frac{\left|\xi_{0}\right|^{2}+2\ell^{2}\left(1-\zeta_{0}^{2}\right)}{\left|\xi_{0}\right|^{2}-2\ell^{2}\left(1-\zeta_{0}^{2}\right)}\right]^{2}.$
(100)

It is easy to see that $\ell=0$ lead the uncertainty relations to the value
obtained when evaluated in terms of the canonical commutation relation
$\left(\left[\hat{x},\hat{p}\right]=i\hbar\text{,
}\hat{p}=-i\hbar\partial_{x}\right)$. Besides, it can be seen that the squeeze
parameter $\zeta_{0}$ ensures that the Heisenberg uncertainty relation
oscillates over time.

This example illustrates how this general procedure can be applied to a range
of problems by simply determining the parameters of the Hamiltonian, then
finding the functions $f$ and $g$. From these functions, we obtain the squeeze
and displacement parameters that, in its turn, modify the uncertainty relation
and probability transition.

##  VII Concluding remarks

In this article, we study the integrals of motion method in a para-Bose
formulation. This approach generalizes the usual canonical commutation
relation. In turn, we obtain a generalization of the usual SVS, which admits a
completeness relation in terms of the Wigner parameter. This relation depends
on a range of values for the Wigner parameter, which does not include that
corresponding to canonical algebra. We also obtain a generalization of the CS
in terms of the even and odd time-independent para-Bose number states. These
states are thoroughly determined in terms of the time-dependent squeeze and
displacement parameters, as well as the Wigner parameter. In the study of the
probability transition, we saw that the displacement parameter has an
additional role, which is a kind of transition parameter by allowing access to
the odd states of the system. Meanwhile, the Wigner parameter has the role of
controlling the “dispersion,” and attenuating the access to the odd states. We
show that the minimization of the Heisenberg uncertainty relation is easily
obtained by taking the real value of the squeeze parameter and that the
squeezing properties can be seen from the standard deviation of the position
and momentum. Taking the coordinate representation of the generalized CS, we
found a quantization condition on the Wigner parameter, analogous to the
quantization of the angular momentum, which arises by imposing that the parity
of the vacuum state is even. This quantization condition also ensures that the
eigenstates of the momentum operator are differentiable at the origin.
Finally, the para-Bose oscillator has been discussed in this framework.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant no.
165/2018), for partial financial support. ASL and FAB acknowledge support from
CNPq (Grant nos. 150601/2021-2, 312104/2018-9). ASP thanks the support of the
Instituto Federal do Pará.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## References

  * (1) Schrödinger E 1926 _Der stetige übergang von der mikro-zur makromechanik_ Naturwissenschaften, Bd. 14, H. 28, 664.
  * (2) Glauber R J 1963 _Coherent and incoherent states of the radiation field_ Phys. Rev. Lett. 131, 2766.
  * (3) Sudarshan E C G 1963 _Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams_ Phys. Lett. 10, 277.
  * (4) Klauder J R 1963 _Continuous-representation Theory. II. Generalized relation between quantum and classical dynamics_ J. Math. Phys. 4, 1058.
  * (5) Scully M O and Zubairy M S 1997 _Quantum Optics_ (Cambridge: Cambridge University Press).
  * (6) Klauder J R and Sudarshan E C 1968 _Fundamentals of Quantum Optics_ (New York: Benjamin).
  * (7) Nielsen M and Chuang I 2000 _Quantum Computation and Quantum Information_ (Cambridge: Cambridge University Press).
  * (8) Klauder J R and Skagerstam B S 1985 _Coherent States, Applications in Physics and Mathematical Physics_ (Singapore: World Scientific).
  * (9) Bachor H A and Ralph T C 2004 _A Guide to Experiments in Quantum Optics_ (Wiley-Vch).
  * (10) Schnabel R 2017 _Squeezed states of light and their applications in laser interferometers_ Phys. Rept. 684, 1.
  * (11) Dey S 2021 _An introductory review on resource theories of generalized nonclassical_ light J. Phys.: Conf. Ser. 2038, 012008.
  * (12) Stoler D 1970 _Equivalence Classes of Minimum Uncertainty Packets_ Phys. Rev. D 1, 3217.
  * (13) Stoler D 1971 _Equivalence Classes of Minimum-Uncertainty Packets. II_ Phys. Rev. D 4, 1925.
  * (14) Slusher R E and Yurke B 1990 _Squeezed light for coherent communications_ J. Light. Tech. 8 466.
  * (15) Caves C M 1981 _Quantum-mechanical noise in an interferometer_ Phys. Rev. D 23, 1693.
  * (16) Ni W-T 1987 _Quantum-mechanical noise in an interferometer: Intrinsic uncertainty versus measurement uncertainty_ Phys. Rev. D 35, 3002.
  * (17) Chua S S Y, Slagmolen B J J, Shaddock D A and McClelland D E 2014 _Quantum squeezed light in gravitational-wave detectors_ Class. Quant. Grav. 31, 183001.
  * (18) Gerry C and Knight P 1983 _Introductory Quantum Optics_ (Cambridge: Cambridge University Press).
  * (19) Malkin I A and Man’ko V I 1979 _Dynamical Symmetries and Coherent States of Quantum Systems_ (Moscow: Nauka).
  * (20) Dodonov V V and Man’ko V I 2003 _Theory of Nonclassical States of Light_ (London: Taylor & Fran-cis Group).
  * (21) Perelomov A M 1986 _Generalized Coherent States and their Applications_ (Verlag: Springer).
  * (22) Gilmore R 1972 _Geometry of symmetrized states_ Ann. Phys. (NY) 74, 391.
  * (23) Ali S T, Antoine J P and Gazeau J P 2000 _Coherent States, Wavelets and Their Generalizations_ (New York: Springer).
  * (24) Yuen H P 1976 _Two-photon coherent states of the radiation field_ Phys. Rev. A 13, 2226.
  * (25) Satyanarayana M V 1985 _Generalized coherent states and generalized squeezed coherent states_ Phys. Rev. D 32, 400.
  * (26) Dey S and Fring A 2012 _Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations_ Phys. Rev. D 86, 064038.
  * (27) Pedram P 2013 _Coherent states in gravitational quantum mechanics_ Int. J. Mod. Phys. D 22, 1350004.
  * (28) Patra P, Saha J P and Biswas K 2022 _Squeezed coherent states for gravitational well in noncommutative space_ Indian J. Phys. 96, 309.
  * (29) Man’ko V I, Marmo G, Sudarshan E C G and Zaccaria F 1997 _f-oscillators and nonlinear coherent states_ Phys. Scr. 55, 528.
  * (30) Dey S 2015 _q-deformed noncommutative cat states and their nonclassical properties_ Phys. Rev. D 91, 044024.
  * (31) Eremin V V and Meldianov A A 2006 _The q-deformed harmonic oscillator, coherent states, and the uncertainty relation_ Theor. Math. Phys. 147, 709.
  * (32) Agarwal G S and Chaturvedi S 1995 _Calogero-Sutherland oscillator: classical behaviour and coherent states_ J. Phys. A: Math and Gen 28, 5747.
  * (33) Brink L, Hansson T H, Konstein S and Vasiliev M A 1993 _The Calogero model – anyonic representation, fermionic extension and supersymmetry_ Nucl. Phys. B 401, 591.
  * (34) Wigner E P 1950 _Do the equations of motion determine the quantum mechanical commutation relations?_ Phys. Rev. 77, 711.
  * (35) Plyushchay M S 1997 _Deformed Heisenberg algebra with reflection_ Nucl. Phys. B 491, 619.
  * (36) Mukunda N, Sudarshan E C G, Sharma J K and Mehta C L 1980 _Representations and properties of para-Bose oscillator operators. I. Energy position and momentum eigenstates_ J. Math. Phys. 21, 2386.
  * (37) Lohe M A and Thilagam A 2004 _Quantum mechanical models in fractional dimensions_ J. Phys. A: Math. Gen. 37, 6181.
  * (38) Sharma J K, Mehta C L, Mukunda N and Sudarshan E C G 1981 _Representation and properties of para-Bose oscillator operators. II. Coherent states and the minimum uncertainty states_ J. Math. Phys. 22, 78.
  * (39) Hartle J B and Taylor J R 1968 _Quantum mechanics of paraparticles_ Phys. Rev. 178, 2043.
  * (40) Green H S 1953 _A generalized method of field quantization_ Phys. Rev. 90, 270.
  * (41) Ohnuki Y and Kamefuchi S 1982 _Quantum Field Theory and Parastatistics_ (New York: Springer).
  * (42) Stoilova N I and Jeugt J V 2010 _Parabosons, parafermions, and explicit representations of infinite-dimensional algebras_ Phys. Atom. Nuclei 73, 533.
  * (43) Greenberg O W and Messiah A M L 1965 _Selection rules for parafields and the absence of para particles in nature_ Phys. Rev. 138, B1155.
  * (44) Kitabayashi T and Yasuè M 2018 _Parafermionic dark matter_ Phys. Rev. D 98, 043504.
  * (45) Bracken A J and Green H S 1973 _Parastatistics and the quark model_ J. Math. Phys. 14, 1784.
  * (46) Hama M, Sawamura M and Suzuki H 1992 _Thermodynamical properties of high order parabosons_ Prog. Theor. Phys. 88, 149.
  * (47) Rodríguez-Walton S, Ávila B J and Rodríguez-Lara B M 2020 _Optical non-Hermitian para-Fermi oscillators_ Phys. Rev. A 101, 043840.
  * (48) Pereira A S and Lemos A S 2021 _Time-dependent coherent squeezed states in a nonunitary approach_ Phys. Lett. A 405, 127428.
  * (49) Meyer H E and Berghe G V 1980 _Generalised Bogoliubov transformation coefficients for para-Bose states_ J. Phys. A: Math. Gen. 13, 161.
  * (50) Jordan T F, Mukunda N and Pepper S V 1963 _Irreducible representations of generalized oscillator operators_ J. Math. Phys. 4, 1089.
  * (51) Sharma J K, Metha C L and Sudarshan E C G 1978 _Para-Bose coherent states_ J. Math. Phys. 19, 2089.
  * (52) Biswas S N and Santhanam T S 1980 _Coherent states of para-Bose oscillators_ J. Austral. Math. Soc. 22, 210.
  * (53) Saxena G M and Mehta C L 1986 _Dynamics of para-Bose coherent states_ J. Math. Phys. 27, 309.
  * (54) Mojaveri B, Dehghani A and Bahrbeig R J 2018 _Nonlinear coherent states of the para-Bose oscillator and their non-classical features_ Eur. Phys. J. Plus 133, 529.
  * (55) Mojaveri B, Dehghani A and Bahrbeig R J 2018 _Excitation on the para-Bose states: nonclassical properties_ Eur. Phys. J. Plus 133, 346.
  * (56) Alderete C H and Vergara L V and Rodríguez-Lara B M 2017 _Nonclassical and semiclassical para-Bose_ states Phys. Rev. A 95, 043835.
  * (57) Bagchi B and Bhaumik D 1997 _Coherent states for parabosons_ J. Phys. A: Math. Gen. 30, L593.
  * (58) Dehghani A, Mojaveri B, Shirin S and Saedi M 2015 _Cat-states in the framework of Wigner-Heisenberg algebra_ Ann. Phys. 362, 659.
  * (59) Calogero F 1971 _Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials_ Jour. Math. Phys. 12, 419.
  * (60) Simons B D, Lee P A and Altshuler B L 1994 _Matrix models, one-dimensional fermions, and quantum chaos_ Phys. Rev. Lett. 72, 64.
  * (61) Kawakami N 1993 _Novel hierarchy of the SU(N) electron models and edge states of fractional quantum Hall effect_ Phys. Rev. Lett. 71, 275.
  * (62) Azuma H and Iso S 1994 _Explicit relation of the quantum Hall effect and the Calogero-Sutherland model_ Phys. Lett. B 331, 107.
  * (63) Ouvry S 2001 _On the relation between the anyon and Calogero models_ Phys. Lett. B 510, 335.
  * (64) Caselle M 1995 _Distribution of transmission eigenvalues in disordered wires_ Phy. Rev. Lett. 74, 2776.
  * (65) Alderete C H, Green A M, Nguyen N H, Zhu Y, Rodríguez-Lara B M and Linke N M 2021 _Experimental realization of para-particle oscillators_ [arXiv:2108.05471 [quant-ph]].
  * (66) Alderete C H and Rodríguez-Lara B M 2017 _Quantum simulation of driven para-Bose oscillators_ Phys. Rev. A 95, 013820.
  * (67) Lewis H R Jr. and Riesenfeld W B 1969 An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. J. Math. Phys. 10, 1458.
  * (68) Dodonov V V, Malkin I A and Man’ko V I 1975 Integrals of the motion, green functions, and coherent states of dynamical systems Int. J. Theor. Phys. 14, 37.
  * (69) Bagrov V G, Gitman D M and Pereira A S 2015 _Coherent states of systems with quadratic Hamiltonians_ Braz. J. Phys. 45, 369.
  * (70) Mark Andrews 1999 _Invariant operators for quadratic Hamiltonians_ Am. J. Phys. 67, 336.
  * (71) Dodonov V V and Man’ko V I 1989 _Evolution of multidimensional systems. Magnetic properties of ideal gases of charged particles. In Invariants and the Evolution of Nonstationary Quantum Systems_ (Proceedings of Lebedev Physics Institute, Volume 183); Markov, M.A., Ed.; Nova Science: Commack, NY, USA; pp. 263.
  * (72) Man’ko V I and Mendes R V 1997 _Time-dependent invariants for Dirac equation and Newton–Wigner position operator_ Phys. Scr. 56, 417.
  * (73) Mostafazadeh A 2001 _Supersymmetric dynamical invariants_ J. Phys. A Math. Gen. 34, 4493.
  * (74) Quesne C 2001 _Completeness of photon-added squeezed vacuum and one-photon states and of photon-added coherent states on a circle_ Phys. Lett. A 288, 241.
  * (75) Chaturvedi S 1996 _A direct proof of the completeness of squeezed odd number states_ Mod. Phys. Lett. A 11, 2805.
  * (76) Glauber R J 1966 _Classical behavior of system of quantum oscillators_ Phys. Lett. 21, 650.
  * (77) Mehta C L and Sudarshan E C G 1966 _Time evolution of coherent states_ Phys. Lett. 22, 574.
  * (78) Robertson H P 1930 _A general formulation of the uncertainty principle and its classical interpretation_ Phys. Rev. 35, 667.
  * (79) Hillery M, O’Connell R F, Scully M O and Wigner E P 1984 _Distribution functions in physics: fundamentals_ Phys. Rep. 106, 121.
  * (80) Coffey W T, Kalmykov Y P and Titov S V 2007 _Solution of the master equation for Wigner’s quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential_ J. Chem. Phys 127, 074502.
  * (81) Olivares S 2012 _Quantum optics in the phase space_ Eur. Phys. J. Special Topics 203, 3.
  * (82) Andrzejewski K 2016 Quantum conformal mechanics emerging from unitary representations of SL(2,$\mathbb{R}$) Annals Phys. 367, 227.
  * (83) Abramowitz M and Stegun I A 1965 _Handbook of mathematical functions_ (New York: Dover).


Using (2) we are able to rewrite the inequalities to conclude that the group
$\Gamma$ is not discrete provided that

$\cos(\alpha)>1-\frac{1}{4\sqrt{3}\operatorname{cosh}^{2}\left(\frac{m}{2}\right)}\quad\mbox{and}\quad\cos(\alpha)\not=1-\frac{\cos(\pi/q)}{4\sqrt{3}\operatorname{cosh}^{2}\left(\frac{m}{2}\right)}$

for some integer $q\geq 3$. ∎

###  9.5. Proof of Proposition 1.9

###### Proof.

The vertical Heisenberg translation element $H$ is given as
$(0,8r^{2}\sqrt{3}\cos^{2}(\theta))$. Substituting this into (1) with
$t=8r^{2}\sqrt{3}\cos^{2}(\theta)$ we have that the group is not discrete if

$\displaystyle\>8r^{2}\sqrt{3}\cos^{2}(\theta)<2\quad$
$\displaystyle\mbox{and}\quad 8r^{2}\sqrt{3}\cos^{2}(\theta)\not=2\cos(\pi/q)$
$\displaystyle\Leftrightarrow\>\cos^{2}(\theta)<\frac{1}{4r^{2}\sqrt{3}}\quad$
$\displaystyle\mbox{and}\quad\cos^{2}(\theta)\not=\frac{\cos(\pi/q)}{4r^{2}\sqrt{3}}$

for some integer $q\geq 3$.
Using (2) we are able to rewrite the inequalities to conclude that the group
$\Gamma$ is not discrete provided that

$\cos(\alpha)>1-\frac{1}{2\sqrt{3}\operatorname{cosh}^{2}\left(\frac{m}{2}\right)}\quad\mbox{and}\quad\cos(\alpha)\not=1-\frac{\cos(\pi/q)}{2\sqrt{3}\operatorname{cosh}^{2}\left(\frac{m}{2}\right)}$

for some integer $q\geq 3$. ∎

##  10\. Acknowledgements

I would like to thank Anna Pratoussevitch and John Parker for their insightful
comments and suggestions throughout this work.

## References

  * [1] H. Brown, R. Bülow, J. Neubüser, H. Wondratscheck and H. Zassenhaus, _Crystallographic groups of four-dimensional Space_ , Wiley, New York, 1978.
  * [2] K. Dekimpe, _Almost-Bieberbach Groups: Affine and Polynomial Structures_ , Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, 1996.
  * [3] W.M. Goldman, _Complex Hyperbolic Geometry_ , Oxford University Press, 1999.
  * [4] S. Hersonsky and F. Paulin, _On the Volumes of Complex Hyperbolic Manifolds_ , Duke Math. J, 84(3):719-737, 1996.
  * [5] A. Monaghan, _Complex Hyperbolic Triangle Groups_ , Ph.D. thesis, University of Liverpool, 2013.
  * [6] A. Monaghan, J.R. Parker and A, Pratoussevitch, _Discreteness of Ultra-Parallel Complex Hyperbolic Triangle Groups of Type $[m_{1},m_{2},0]$_, Journal of the LMS, 100:545-567, 2019.
  * [7] J.R. Parker, _On Ford isometric spheres in complex hyperbolic space_ , Math. Proc. Camb. Phil. Soc. 115:501–512, 1994.
  * [8] J.R. Parker, _On the Volumes of Cusped, Complex Hyperbolic Manifolds and Orbifolds_ , Duke Math. J, 94:433–464, 1998.
  * [9] J.R. Parker, _Notes on Complex Hyperbolic Geometry_ , lecture notes, 2010.
  * [10] J.R. Parker and J. Paupert _Unfaithful Complex Hyperbolic Triangle Groups, II: Higher Order Reflections_ , Pacific Journal of Mathematics, 239:357–389, 2009.
  * [11] S. Povall, _Ultra-Parallel Complex Hyperbolic Triangle Groups_ , Ph.D. thesis, University of Liverpool, 2019.
  * [12] S. Povall and A. Pratoussevitch, _Complex Hyperbolic Triangle Groups of Type $[m,m,0;3,3,2]$_, Conformal Geometry and Dynamics, 24:51–67, 2020.
  * [13] A. Pratoussevitch, _Traces in Complex Hyperbolic Triangle Groups_ , Geometriae Dedicata 111:159–185, 2005.
  * [14] R.E. Schwartz, _Complex hyperbolic triangle groups_ , Proceedings of the ICM (Beijing, 2002), Vol II, Higher Ed. Press, 339–349, 2002.
  * [15] J. Wyss-Gallifent, _Complex Hyperbolic Triangle Groups_ , Ph.D. thesis, University of Maryland, 2000.