# Bi-Equivariant Fibrations

Pavel S. Gevorgyan  Moscow Pedagogical State University<EMAIL_ADDRESS>
###### Abstract.

The lifting problem for continuous bi-equivariant maps and bi-equivariant
covering homotopies is considered, which leads to the notion of a bi-
equivariant fibration. An intrinsic characteristic of a bi-equivariant
Hurewicz fibration is obtained. Theorems concerning a relationship between bi-
equivariant fibrations and fibrations generated by them are proved.

###### Key words and phrases:

binary $G$-space, distributive action, bi-equivariant covering homotopy, bi-
equivariant fibration, $H$-fixed point, orbit space

###### 1991 Mathematics Subject Classification:

54H15, 55R91

##  1\. Introduction

The foundations of bi-equivariant topology were laid in [4], where the notions
of a binary $G$-space and of a bi-equivariant map were introduced. The space
$H_{2}(X)$ of all continuous invertible binary operations on a locally compact
locally connected space $X$ with left multiplication
$AB(x,x^{\prime})=A(x,B(x,x^{\prime}))$ for $A,B\in H_{2}(X)$ and
$x,x^{\prime}\in X$ is a topological group which acts binarily on the space
$X$. If a topological group $G$ acts binarily and effectively on $X$, then $G$
is a subgroup of $H_{2}(X)$ [2]. All binary $G$-spaces and bi-equivariant maps
form a category, which is a natural extension of the category of $G$-spaces
and equivariant maps. Set-theoretic questions related to bi-equivariant
topology were also considered in [2]–[7].

This paper is devoted to the important problems of lifting continuous bi-
equivariant maps and bi-equivariant covering homotopies, which lead to bi-
equivariant fibrations.

To study bi-equivariant Hurewicz fibrations, we introduce the notion of a bi-
equivariant covering function and prove that a map $p\colon E\to B$ is a bi-
equivariant covering homotopy with respect to any binary $G$-space $X$ if and
only if $p$ has a bi-equivariant covering function. This is an intrinsic
characteristic of a bi-equivariant Hurewicz fibration, because the definition
of a bi-equivariant covering function for a map $p\colon E\to B$ is not
related to a binary $G$-space $X$.

The property of being a bi-equivariant covering homotopy is preserved under
the passage to closed subgroups of $G$; i.e., a bi-equivariant Hurewicz
$G$-fibration $p\colon E\to B$ is also a bi-equivariant $H$-fibration.

In the case where $E$ and $B$ are distributive binary $G$-spaces, a bi-
equivariant surjective map $p\colon E\to B$ induces a bi-equivariant map
$p^{H}\colon E^{H}\to B^{H}$ of the spaces of $H$-fixed points, which is a bi-
equivariant Hurewicz $G$-fibration if so is $p\colon E\to B$.

In the case where $G$ is a compact Abelian Lie group and $E$ and $B$ are
distributive binary $G$-spaces, a bi-equivariant Hurewicz fibration $p\colon
E\to B$ induces a Hurewicz fibration $p^{*}\colon E^{*}\to B^{*}$ of the orbit
spaces.

##  2\. Auxiliary Definitions and Results

###  2.1. Binary $G$-Spaces and bi-equivariant Maps

Let $G$ be any topological group, and let $X$ be any topological space.

A continuous map $\alpha\colon G\times X^{2}\to X$ is called a binary action
of the topological group $G$ on the space $X$ if, for any $g,h\in G$ and
$x_{1},x_{2}\in X$,

(1) $gh(x,x^{\prime})=g(x,h(x,x^{\prime})),$ (2) $e(x,x^{\prime})=x^{\prime},$

where $e$ is the identity element of $G$ and
$g(x,x^{\prime})=\alpha(g,x,x^{\prime})$.

A triple $(G,X,\alpha)$ consisting of a space $X$, a group $G$, and a fixed
binary action $\alpha$ of $G$ on $X$ is called a binary $G$-space.

Let $(G,X,\alpha)$ and $(G,Y,\beta)$ be binary $G$-spaces. A continuous map
$f\colon X\to Y$ is said to be bi-equivariant, or _$G$ -bi-equivariant_, if

$f(\alpha(g,x,x^{\prime}))=\beta(g,f(x),f(x^{\prime})),$

or

$f(g(x,x^{\prime}))=g(f(x),f(x^{\prime})),$

for all $g\in G$ and $x,x^{\prime}\in X$.

All binary $G$-spaces and bi-equivariant maps form a category, which we denote
by Bi-$G$-$Top$.

Note that, given any $G$-space $(X,G,\alpha)$, the unary action $\alpha$ on
$X$ generates the binary action $\overline{\alpha}$ defined by

(3) $\overline{\alpha}(g,x,x^{\prime})=\alpha(g,x^{\prime})\quad\text{or}\quad
g(x,x^{\prime})=gx^{\prime}$

for all $g\in G$ and $x,x^{\prime}\in X$, and any equivariant map
$f:(G,X,\alpha)\to(G,Y,\beta)$ between $G$-spaces $X$ and $Y$ is bi-
equivariant with respect to the action (3). Indeed,

$f(\overline{\alpha}(g,x,x^{\prime}))=f(\alpha(g,x^{\prime}))=\beta(g,f(x^{\prime}))=\overline{\beta}(g,f(x),f(x^{\prime})).$

Thus, the category Bi-$G$-$Top$ can be considered as a natural extension of
the category $G$-$Top$ of all $G$-spaces and equivariant maps.

Let $H$ be a subgroup of a group $G$. Then any binary $G$-space is also a
binary $H$-space, and any $G$-bi-equivariant map is an $H$-bi-equivariant map.
Thus, there exists a natural covariant functor from the category Bi-$G$-TOP to
the category Bi-$H$-TOP.

Let $X$ be a binary $G$-space, and let $A$ be its subset. The set
$G(A,A)=\\{g(a,a^{\prime});g\in G,a,a^{\prime}\in A\\}$ is called the _binary
saturation_ of $A$. A subset $A$ of a binary $G$-space $X$ is said to be _bi-
invariant_ , or _$G$ -bi-invariant_, if $A$ coincides with its binary
saturation: $G(A,A)=A$. A bi-invariant subset $A\subset X$ is itself a binary
$G$-space; it is called a _binary $G$-subspace_ of the space $X$.

A binary $G$-space $X$ is said to be distributive if, for any
$x,x^{\prime},x^{\prime\prime}\in X$ and $g,h\in G$, we have

(4)
$g(h(x,x^{\prime}),h(x,x^{\prime\prime}))=h(x,g(x^{\prime},x^{\prime\prime})).$

The class of distributive binary $G$-spaces plays an important role in the
theory of binary $G$-spaces. This is explained, in particular, by the special
role of distributive subgroups of the group $H_{2}(X)$ of all continuous
invertible binary operations on $X$. For example, any topological group is a
distributive subgroup of the group of invertible binary operations on some
space [3]. This statement is a binary topological analogue of Cayley’s
classical theorem on the representation of any finite group by unary
operations (permutations).

###  2.2. The space of $H$-fixed points of a binary $G$-space

Let $X$ be a binary $G$-space, and let $H$ be a subgroup of $G$. The set

$X^{H}=\\{x^{\prime}\in X;\quad h(x,x^{\prime})=x^{\prime}\ \text{for all}\
h\in H,\ x\in X\\}$

is called the _space of $H$-fixed points_ of the binary $G$-space $X$. This
set is not generally $G$-bi-invariant. However, the following assertion is
valid.

######  Proposition 1.

If $X$ is a distributive binary $G$-space, then $X^{H}$ is a $G$-bi-invariant
subset and, therefore, a binary $G$-subspace.

###### Proof.

Let $x^{\prime},x^{\prime\prime}\in X^{H}$ be any $H$-fixed points. Then,
since the binary action on $X$ is distributive, it follows that, for any $g\in
G$, $h\in H$ and $x\in X$, we have

$h(x,g(x^{\prime},x^{\prime\prime}))=g(h(x,x^{\prime}),h(x,x^{\prime\prime}))=g(x^{\prime},x^{\prime\prime}),$

i.e., $g(x^{\prime},x^{\prime\prime})\in X^{H}$. Thus, $G(X^{H},X^{H})=X^{H}$.
∎

######  Proposition 2.

If $X$ and $Y$ are binary $G$-spaces and $f\colon X\to Y$ is a surjective bi-
equivariant map, then $f(X^{H})\subset Y^{H}$.

###### Proof.

Indeed, for any $x^{\prime}\in X^{H}$, we have

$h(y,f(x^{\prime}))=h(f(x),f(x^{\prime}))=f(h(x,x^{\prime}))=f(x^{\prime}),$

where $y\in Y$ is any element and $x\in X$ is a preimage of $y$. Therefore,
$f(x^{\prime})\in Y^{H}$. ∎

We denote the restriction of a surjective bi-equivariant map $f\colon X\to Y$
to $X^{H}$ by $f^{H}\colon X^{H}\to Y^{H}$. Thus, if $X$ is a distributive
binary $G$-space, then $f^{H}\colon X^{H}\to Y^{H}$ is a bi-equivariant map of
binary $G$-spaces.

###  2.3. The orbit space of a distributive binary $G$-space

Let $X$ be a binary $G$-space. The orbit of a point $x\in X$ is the minimal
bi-invariant subset $[x]$ of $X$ containing the point $x$. Obviously, $x\in
G(x,x)\subset[x]$ for all $x\in X$. Therefore, if $G(x,x)$ is a bi-invariant
set, then $G(x,x)=[x]$.

As is known, the set $G(x,x)$ is not generally bi-invariant [7, Example 2].
However, in distributive binary $G$-spaces the sets $G(x,x)$ are bi-invariant,
and hence $[x]=G(x,x)$ [2, Theorem 9]. Moreover, the orbits of a distributive
binary $G$-space either are disjoint or coincide [7, Proposition 6];
therefore, the space $X$ is partitioned into disjoint classes. We denote the
corresponding quotient set by $X|G$.

Let $\pi=\pi_{X}:X\to X|G$ be the natural quotient map which sends each point
$x\in X$ to its orbit $[x]$. The quotient topology on $X|G$ is defined in the
standard way. The space thus obtained is called the _orbit space_ of the
distributive binary $G$-space $X$.

If $X$ and $Y$ are distributive binary $G$-spaces, then any bi-equivariant map
$f:X\to Y$ generates a map $f^{*}:X|G\to Y|G$ of the orbit spaces, which is
defined by $f^{*}([x])=[f(x)]$. This map is well defined, because it does not
depend on the choice of representatives of orbits:
$f^{*}([g(x,x)])=[f(g(x,x))]=[g(f(x),f(x))]=[f(x)]=f^{*}([x])$ for any $g\in
G$.

All notions, definitions, and results used in the paper without references, as
well as all those mentioned above, can be found in [2]–[7].

###  2.4. The bi-equivariant covering homotopy property and bi-equivariant
Hurewicz fibrations

Important problems of algebraic topology are the problem of _lifting_
continuous maps and the _covering homotopy problem_ , which is dual to the
problem of extending homotopies. The covering homotopy problem leads to the
notion of a fibration. We study these problems in the category Bi-$G$-TOP of
binary $G$-spaces and bi-equivariant maps.

Let $G$ be a topological group. Suppose given binary $G$-spaces $E$, $B$ and
$X$ and bi-equivariant maps $p\colon E\to B$ and $f\colon X\to B$. The
_problem of lifting_ the bi-equivariant map $f$ to $E$ consists in determining
whether there exists a continuous bi-equivariant map $\tilde{f}:X\to E$ such
that $f=p\circ\tilde{f}$; this map is denoted by dashed arrow in the diagram

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\tilde{f}}$$\textstyle{B.}$

To turn this problem into a well-posed problem of binary $G$-homotopy
category, we need the counterpart of the _homotopy extension property_ , which
is called the _bi-equivariant covering homotopy property_. We say that a bi-
equivariant map $p:E\to B$ has the _bi-equivariant covering homotopy property_
with respect to a binary $G$-space $X$ if, for any bi-equivariant maps
$\tilde{f}:X\to E$ and $F:X\times I\to B$ such that $F\circ
i_{0}=p\circ\tilde{f}$, where $i_{0}:X\to X\times I$ is the embedding defined
by $i_{0}(x)=(x,0)$, $x\in X$, there exists a bi-equivariant map
$\tilde{F}:X\times I\to E$ for which the diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\scriptstyle{i_{0}}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{X\times
I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\scriptstyle{\tilde{F}}$$\textstyle{B}$

is commutative, i.e., $p\circ\tilde{F}=F$ and $\tilde{F}\circ
i_{0}=\tilde{f}$.

Obviously, if $p:E\to B$ has the bi-equivariant covering homotopy property
with respect to a binary $G$-space $X$ and $f,g:X\to B$ are $G$-homotopic bi-
equivariant maps, then $f$ lifts to $E$ if and only if so does $g$. Thus, the
existence of a lifting of a bi-equivariant map $f:X\to B$ is a property of a
binary $G$-homotopy class of this map.

The bi-equivariant covering homotopy property leads to the notion of a bi-
equivariant fibration. A bi-equivariant map $p:E\to B$ is called a _bi-
equivariant Hurewicz fibration_ , or a _bi-equivariant Hurewicz
$G$-fibration_, if $p$ has the bi-equivariant covering homotopy property with
respect to any binary $G$-space $X$. In this case, the binary $G$-space $E$ is
called the _space of a bi-equivariant fibration_ , and $B$ is the _base_ of
this fibration.

##  3\. Covering Binary $G$-Functions
and an Intrinsic Characteristic of bi-equivariant Fibrations

Let $G$ be a compact topological group. Suppose given binary $G$-spaces $E$
and $B$ and a bi-equivariant map $p:E\to B$.

For the path space $B^{I}$, we define a map $G\times B^{I}\times B^{I}\to
B^{I}$ by

(5) $g(\alpha,\alpha^{\prime})(t)=g(\alpha(t),\alpha^{\prime}(t)),$

where $g\in G$, $\alpha,\alpha^{\prime}\in B^{I}$, and $t\in I$.

######  Proposition 3.

The map (5) defines a continuous binary action of the group $G$ on the path
space $B^{I}$.

###### Proof.

The continuity of the map (5) follows from that of $\alpha$ and
$\alpha^{\prime}$ and of the binary action of $G$ on $B$.

Let us check that the action (5) is binary. For any $g,h\in G$,
$\alpha,\alpha^{\prime}\in B^{I}$, and $t,t^{\prime}\in I$, we have

(1)
$e(\alpha,\alpha^{\prime})(t)=g(\alpha(t),\alpha^{\prime}(t))=\alpha^{\prime}(t)$,
i.e., $e(\alpha,\alpha^{\prime})=\alpha^{\prime}$;

$(2)\
gh(\alpha,\alpha^{\prime})(t)=gh(\alpha(t),\alpha^{\prime}(t))=g(\alpha(t),h(\alpha(t),\alpha^{\prime}(t)))=\\\
=g(\alpha(t),h(\alpha,\alpha^{\prime})(t))=g(\alpha,h(\alpha,\alpha^{\prime}))(t),$

i.e., $gh(\alpha,\alpha^{\prime})=g(\alpha,h(\alpha,\alpha^{\prime}))$. ∎

Since $E$ and $B^{I}$ are binary $G$-spaces, it follows that the product
$E\times B^{I}$ is a binary $G$-space with coordinatewise binary action.
Consider the subspace $\Delta\subset E\times B^{I}$ defined by

$\Delta=\\{(e,\alpha)\in E\times B^{I};\ \alpha(0)=p(e)\\}.$

######  Proposition 4.

The set $\Delta$ is a bi-invariant subspace of the binary $G$-space $E\times
B^{I}$.

###### Proof.

Let $(e,\alpha),(e^{\prime},\alpha^{\prime})\in\Delta$ be any elements, i.e.,
$\alpha(0)=p(e)$ and $\alpha^{\prime}(0)=p(e^{\prime})$. Then
$g((e,\alpha),(e^{\prime},\alpha^{\prime}))=(g(e,e^{\prime}),g(\alpha,\alpha^{\prime}))$
and

$g(\alpha,\alpha^{\prime})(0)=g(\alpha(0),\alpha^{\prime}(0))=g(p(e),p(e^{\prime}))=p(g(e,e^{\prime})).$

Therefore, $g((e,\alpha),(e^{\prime},\alpha^{\prime}))\in\Delta$ for any $g\in
G$. ∎

######  Definition 1.

A bi-equivariant map $\lambda:\Delta\to E^{I}$ satisfying the conditions

(6)
$\lambda(e,\alpha)(0)=e\quad\text{and}\quad[p\circ\lambda(e,\alpha)](t)=\alpha(t)$

is called a _bi-equivariant covering function_ , or a _covering bi-equivariant
$G$-function_, for $p$.

The following theorem describes a relationship between bi-equivariant Hurewicz
fibrations and bi-equivariant covering functions.

######  Theorem 1.

A bi-equivariant map $p:E\to B$ is a bi-equivariant Hurewicz fibration if and
only if $p$ has a bi-equivariant covering function.

###### Proof.

Let $\lambda:\Delta\to E^{I}$ be a bi-equivariant covering function for $p$.
Consider any binary $G$-space $X$, bi-equivariant map $\tilde{f}:X\times 0\to
E$, and bi-equivariant homotopy $F:X\times I\to B$ for which
$F(x,0)=p(\tilde{f}(x,0))$.

Note that, for any $x\in X$, the formula $F_{x}(t)=F(x,t)$ defines a path
$F_{x}\in B^{I}$. It follows from the bi-equivariance of $F$ that

(7) $F_{g(x,x^{\prime})}=g(F_{x},F_{x^{\prime}}).$

Indeed,

$F_{g(x,x^{\prime})}(t)=F(g(x,x^{\prime}),t)=g(F(x,t),F(x^{\prime},t))=g(F_{x}(t),F_{x^{\prime}}(t))=g(F_{x},F_{x^{\prime}})(t).$

Now consider the homotopy $\tilde{F}:X\times I\to E$ defined by

(8) $\tilde{F}(x,t)=\lambda(\tilde{f}(x,0),F_{x})(t).$

Let us prove that $\tilde{F}$ is the required bi-equivariant covering
homotopy.

The bi-equivariance of the homotopy $\tilde{F}$ follows from that of the maps
$\lambda$, $\tilde{f}$, and $F$ and relations (5), (7), and (8):

$\tilde{F}(g(x,x^{\prime}),t)=\lambda(\tilde{f}(g(x,x^{\prime}),0),F_{g(x,x^{\prime})})(t)=\lambda(g(\tilde{f}(x,0),\tilde{f}(x^{\prime},0)),g(F_{x},F_{x}^{\prime}))(t)=\\\
=\lambda(g((\tilde{f}(x,0),F_{x}),(\tilde{f}(x^{\prime},0),F_{x}^{\prime})))(t)=g(\lambda(\tilde{f}(x,0),F_{x}),\lambda(\tilde{f}(x^{\prime},0),F_{x}^{\prime}))(t)=\\\
=g(\lambda(\tilde{f}(x,0),F_{x})(t),\lambda(\tilde{f}(x^{\prime},0),F_{x}^{\prime})(t))=g(\tilde{F}(x,t),\tilde{F}(x^{\prime},t)).$

According to conditions (6), we have

$\tilde{F}(x,0)=\lambda(\tilde{f}(x,0),F_{x})(0)=\tilde{f}(x,0),$
$(p\circ\tilde{F})(x,t)=p(\tilde{F}(x,t))=p(\lambda(\tilde{f}(x,0),F_{x})(t))=[p\circ\lambda(\tilde{f}(x,0),F_{x})](t)=F_{x}(t)=F(x,t).$

Therefore, $\tilde{F}$ is a covering homotopy.

Now suppose that $p:E\to B$ is a bi-equivariant Hurewicz fibration. Consider
the binary $G$-space $X=\Delta$ and the maps

$\tilde{f}:\Delta\times 0\to E\quad\text{and}\quad F:\Delta\times I\to B$

defined by

(9) $\tilde{f}[(e,\alpha),0]=e\quad\text{and}\quad F[(e,\alpha),t]=\alpha(t).$

The maps $\tilde{f}$ and $F$ are bi-equivariant. Indeed, for any $g\in G$,
$(e,\alpha),(e^{\prime},\alpha^{\prime})\in\Delta\subset E\times B^{I}$, and
$t\in I$, we have

$\tilde{f}[g((e,\alpha),(e^{\prime},\alpha^{\prime})),0]=\tilde{f}[(g(e,e^{\prime}),g(\alpha,\alpha^{\prime})),0]=g(e,e^{\prime})=g(\tilde{f}[(e,\alpha),0],\tilde{f}[(e^{\prime},\alpha^{\prime}),0]),$
$F[g((e,\alpha),(e^{\prime},\alpha^{\prime})),t]=F[(g(e,e^{\prime}),g(\alpha,\alpha^{\prime})),t]=g(\alpha,\alpha^{\prime})(t)=\\\
=g(\alpha(t),\alpha^{\prime}(t))=g(F[(e,\alpha),t],F[(e^{\prime},\alpha^{\prime}),t]).$

Note that $F:\Delta\times I\to B$ is a bi-equivariant homotopy of the map
$p\circ\tilde{f}:\Delta\times 0\to B$. Indeed,

$F[(e,\alpha),0]=\alpha(0)=p(e)=p(\tilde{f}[(e,\alpha),0])=(p\circ\tilde{f})[(e,\alpha),0].$

Hence there exists a bi-equivariant covering homotopy $\tilde{F}:\Delta\times
I\to E$ for $F$, i.e.,

(10) $\tilde{F}[(e,\alpha),0]=\tilde{f}[(e,\alpha),0]\quad\text{and}\quad
p\circ\tilde{F}=F.$

Now consider the map $\lambda:\Delta\to E^{I}$ defined by

(11) $\lambda(e,\alpha)(t)=\tilde{F}[(e,\alpha),t].$

Let us prove that $\lambda$ is a covering bi-equivariant $G$-function for $p$.
The bi-equivariance of $\lambda$ follows from that of the covering homotopy
$\tilde{F}$:

$\lambda(g((e,\alpha),(e^{\prime},\alpha^{\prime})))(t)=\tilde{F}[g((e,\alpha),(e^{\prime},\alpha^{\prime})),t]=\\\
=g(\tilde{F}[(e,\alpha),t],\tilde{F}[(e^{\prime},\alpha^{\prime}),t])=g(\lambda(e,\alpha)(t),\lambda(e^{\prime},\alpha^{\prime})(t)).$

Conditions (6) in Definition 1 also hold. By virtue of (9), (10), and (11), we
have

$\lambda(e,\alpha)(0)=\tilde{F}[(e,\alpha),0]=\tilde{f}[(e,\alpha),0]=e,$
$[p\circ\lambda(e,\alpha)](t)=p(\lambda(e,\alpha)(t))=p(\tilde{F}[(e,\alpha),t])=F[(e,\alpha),t]=\alpha(t).$

This completes the proof of the theorem. ∎

Note that the last theorem gives an intrinsic characteristic of a bi-
equivariant Hurewicz fibration, because Definition 1 of a bi-equivariant
covering function does not involve any outer space $X$.

Applying Theorem 1 to the trivial binary action, we obtain the following
result.

######  Corollary 1.

A continuous map $p:E\to B$ is a Hurewicz fibration if and only if $p$ has a
covering function.

##  4\. Fibrations Generated by bi-equivariant $G$-Fibrations

Let $H$ be a closed subgroup of a compact group $G$. Since all binary
$G$-spaces are also binary $H$-spaces, and all $G$-bi-equivariant maps are
$H$-bi-equivariant maps, there arises the natural question of whether the
property of being a bi-equivariant Hurewicz foliation is preserved under the
passage to a closed subgroup $H$.

######  Theorem 2.

Let $p:E\to B$ be a bi-equivariant Hurewicz $G$-fibration. Then, for any
closed subgroup $H$ of the compact group $G$, the map $p:E\to B$ is also a bi-
equivariant Hurewicz $H$-fibration.

###### Proof.

By virtue of Theorem 1, there exists a covering bi-equivariant $G$-function
$\lambda:\Delta\to E^{I}$ for $p$. Since $\lambda:\Delta\to E^{I}$ is also a
bi-equivariant $H$-map, it follows that $\lambda$ is a covering bi-equivariant
$H$-function for the bi-equivariant $H$-map $p$. Therefore, $p:E\to B$ is a
bi-equivariant Hurewicz $H$-fibration by virtue of the same Theorem 1. ∎

Under certain constraints, the property of being a bi-equivariant Hurewicz
$G$-fibration is preserved under the passage to binary $G$-subspaces of
$H$-fixed points.

######  Theorem 3.

Let $E$ and $B$ be a distributive binary $G$-space, and let $p:E\to B$ be a
surjective bi-equivariant Hurewicz $G$-fibration. Then, for any closed
subgroup $H$ of the compact group $G$, the induced bi-equivariant $G$-map
$p^{H}:E^{H}\to B^{H}$ between the spaces of $H$-fixed points is a bi-
equivariant Hurewicz $G$-fibration.

###### Proof.

The bi-invariance of the set of $H$-fixed points of a distributive binary
$G$-space and the preservation of $H$-fixed points under a surjective bi-
equivariant map were proved in Propositions 1 and 2.

Let $\lambda:\Delta\to E^{I}$ be a covering bi-equivariant $G$-function for
$p$, whose existence follows from Theorem 1.

Consider the set

$\Delta^{H}=\\{(\dot{e},\dot{\alpha})\in E^{H}\times(B^{H})^{I};\
\dot{\alpha}(0)=p^{H}(\dot{e})\\}.$

Let us prove the existence of a covering bi-equivariant $G$-function
$\lambda^{H}:\Delta^{H}\to(E^{H})^{I}$ for a bi-equivariant map
$p^{H}:E^{H}\to B^{H}$.

Since $\Delta^{H}\subset\Delta$, we can set $\lambda^{H}=\lambda|\Delta^{H}$.
It is easy to see that $\lambda^{H}$ takes the set $\Delta^{H}$ to
$(E^{H})^{I}$ and that $\lambda^{H}:\Delta^{H}\to(E^{H})^{I}$ is a covering
bi-equivariant $G$-function for the map $p^{H}:E^{H}\to B^{H}$. Therefore, by
virtue of Theorem 1, $p^{H}:E^{H}\to B^{H}$ is a bi-equivariant Hurewicz
$G$-fibration. ∎

In the case of distributive binary $G$-spaces, the property of being a binary
Hurewicz $G$-fibration is also preserved under the passage to orbit spaces.

######  Theorem 4.

Let $G$ be a compact Abelian Lie group, and let $E$ and $B$ be distributive
binary $G$-spaces. If $p:E\to B$ is a bi-equivariant Hurewicz $G$-fibration,
then the induced map $p^{*}:E|G\to B|G$ of orbit spaces is a Hurewicz
fibration.

###### Proof.

Let $p:E\to B$ be a bi-equivariant Hurewicz $G$-fibration. By Theorem 1 there
exists a covering bi-equivariant $G$-function $\lambda:\Delta\to E^{I}$ for
$p$.

Consider the set

$\Delta_{G}=\\{(e^{*},\alpha^{*})\in E|G\times(B|G)^{I};\
\alpha^{*}(0)=p^{*}(e^{*})\\}.$

Let us prove that there exists a covering function
$\lambda^{*}:\Delta_{G}\to(E|G)^{I}$ for the map $p^{*}:E|G\to B|G$.

Since $B$ is a distributive binary $G$-space, it follows that its orbits have
the form $G(x,x)$ for any $x\in B$. Note that $B|G$ can also be treated as the
orbit space of the $G$-space $B$ with the action $gx=g(x,x)$, $g\in G$, $x\in
B$. This formula indeed defines an action of the group $G$ on $B$, because the
group $G$ is commutative and the binary action of the group $G$ on $B$ is
distributive.

Note that any path $\alpha^{*}:I\to B|G$ can be lifted to $B$ ([1, Theorem
6.2]), i.e., there exists a path $\alpha:I\to B$ such that
$\pi_{B}\circ\alpha=\alpha^{*}$, where $\pi_{B}:B\to B|G$ is the orbit
projection. Now we define the required map
$\lambda^{*}:\Delta_{G}\to(E|G)^{I}$ by

$\lambda^{*}(e^{*},\alpha^{*})(t)=(\lambda(e,\alpha)(t))^{*}.$

The map $\lambda^{*}$ is well defined, and it is a covering function for
$p^{*}:E|G\to B|G$. Therefore, by Corollary 1, $p^{*}:E|G\to B|G$ is a
Hurewicz fibration. ∎

## References

  * [1] G. E. Bredon, Introduction to Compact Transformation Groups. Academic Press, New York, 1972.
  * [2] P. S. Gevorgyan, “Groups of binary operations and binary G-spaces,” _Topology and its Applications, 201, 2016, 18–28._
  * [3] P. S. Gevorgyan, “Groups of Invertible Binary Operations of a Topological Space,” _Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 2018, 53, 16–20_.
  * [4] P. S. Gevorgyan, “On Binary G-Spaces,” _Mathematical Notes, 96:4, 2014, 127–129._
  * [5] P. S. Gevorgyan, S. D. Iliadis, “Groups of generalized isotopies and generalized G-spaces,” _Matematicki Vesnik, 70:2 (2018), 110–119._
  * [6] P. S. Gevorgyan, R. Jimenez, “On equivariant fibrations of G-CW-complexes,” _Sbornik: Mathematics, 2019, 210:10, 1428–1433_.
  * [7] P. S. Gevorgyan, A. A. Nazaryan, “On Orbits and Bi-invariant Subsets of Binary G-Spaces,” _Mathematical Notes, 109:1 (2021), 38–45._
  * [8] E. H. Spanier, Algebraic Topology. M.: Mir, 1971.


# Quantum theory of a harmonic oscillator in a time dependent noncommutative
background

Manjari Duttaa<EMAIL_ADDRESS><EMAIL_ADDRESS>Shreemoyee Ganguly b<EMAIL_ADDRESS>Sunandan Gangopadhyay c
a,c Department of Astrophysics and High Energy physics
<EMAIL_ADDRESS><EMAIL_ADDRESS>
S.N. Bose National Centre for Basic Sciences
JD Block, Sector III, Salt Lake, Kolkata 700106, India
b Department of Basic Science and Humanities,
University of Engineering and Management (UEM),
B/5, Plot No.III, Action Area-III, Newtown, Kolkata 700156

###### Abstract

This work explores the behaviour of a noncommutative harmonic oscillator in a
time-dependent background, as previously investigated by Dey et al. [1].
Specifically, we examine the system when expressed in terms of commutative
variables, utilizing a generalized form of the standard Bopp-shift relations
recently introduced by [2]. We solved the time dependent system and obtained
the analytical form of the eigenfunction using Lewis’ method of invariants,
which is associated with the Ermakov-Pinney equation, a non-linear
differential equation. We then explicitly provided the exact analytical
solution set for the Ermakov-Pinney equation. Then, we computed the dynamics
of the energy expectation value analytically and explored their graphical
representations for various solution sets of the Ermakov-Pinney equation,
associated with a particular choice of quantum number. Finally, we determined
the generalized form of the uncertainty equality relations among the operators
for both commutative and noncommutative cases. Expectedly, our study is
consistent with the findings in [1], specifically in a particular limit where
the coordinate mapping relations reduce to the standard Bopp-shift relations.

## 1 Introduction

The concept of noncommutative (NC) spacetime as well as the discreteness of
spacetime structure seems to be an essential requirement to unify the general
theory of relativity and the laws of quantum mechanics. When measuring the
small distance at around Planck scale, both the momentum and the energy of the
probing particle become very high, in accordance with Heisenberg’s uncertainty
principle [3]. Consequently, this leads to the collapse of the spacetime
manifold, ultimately causing the breakdown of the general theory of relativity
as well as the laws of gravitation at this extremely small length scale.
Therefore, in order to unify both the general theory of relativity and quantum
field theory, it is necessary to employ a minimum observable length, beyond
which measurement is impossible in the spacetime manifold.
The idea of NC quantum theory was first historically introduced by Heisenberg
(in the late 1930, he wrote to Pierls in a letter [4]). Finally, the idea is
formalised in the pioneering work [5] by Snyder and the study of quantum
mechanical systems in NC spaces has captivated theoretical physicists. In that
work, by providing an example of a Lorentz invariant discrete spacetime, it is
shown that the usual assumption of continuuum spacetime is not required by
Lorentz invariance. Immediately after this work, an extension to this
treatment is also performed by [6] for the case of curved spacetime
(specifically de Sitter space). Although, at that time that idea is largely
ignored, the NC nature of spacetime is later strongly supported by string
theory [9, 8, 7] which is one of the current theories of quantum gravity. The
theory comes up with a concept of a finite length $l_{s}$ which is actually
the length of the strings and measures the short distance structure. Hence, it
is not possible to observe the distance which is even smaller than $l_{s}$. It
was also shown in the seminal work [10] that at a certain low energy limit,
string theory can be treated as an effective quantum field theory in NC
spacetime [11, 12]. However, this is a fact that, almost all the theories of
quantum gravity implies the existence of minimum measurable length as well as
the discreteness of spacetime which can resist the gravitational collapse at
microscopic level. Hence, the consideration of NC spacetime can also ensure
gravitational stability [13, 14]. Apart from these, the problem of
understanding quantum spacetime is also approached by loop quantum gravity
[15, 16]. As a result, several studies on quantum mechanical systems in NC
spaces have been conducted and reported in the literature [17]-[27].

The simplest model of NC spacetime is a two-dimensional quantum mechanical
space, where the standard set of commutation relations between the canonical
coordinates is replaced by NC commutation relations $[X,Y]=i\theta$, where
$\theta$ is a positive real constant. Studies involving NC spaces may also
include the NC commutation relations $[P_{x},P_{y}]=i\Omega$, where $\Omega$
is a real constant. However, more interesting structural form of $\theta$ can
be found as the function of momenta and coordinates [28, 29, 30, 31, 32, 33]
in the literature. In the literature on noncommutativity, it has been found to
be convenient to use the standard Bopp-shift [34] relations to map a system in
NC space in terms of commutative variables. However, in a recent communication
[2], a modified version of the standard Bopp-shift relations have been
proposed. The main difference between two types of coordinate mapping
relations lies in the introduced relations in [2], where the commutation
relations between the spatial coordinates and the momentum coordinates simply
obey the phase-space commutation relation in the ordinary quantum mechanics
i.e. $[X_{j},P_{j}]=i$ ; where $j\,\in\,[1,2]$. This contrasts with the
standard Bopp-shift relations, as the commutator $[X_{j},P_{j}]$ is a function
of the NC parameters $\theta$ and $\Omega$. Another distinction arises from
the fact that, although in both the standard version and the modified version
of the coordinate mapping relations, the NC parameters $\theta,\Omega$ are
positive, real constants, the second version imposes the condition
$\theta\Omega\,<\,0$. Consequently, it seems to be interesting to study a
quantum mechanical system within the NC framework proposed in [2]. In contrast
to the study mentioned above, where the NC parameters were set to be
constants, we adopt their coordinate mapping relations by allowing the NC
parameters to vary with respect to time in our work. In fact, the time-varying
NC parameters can be thought to arise from the renormalization group flow of
Newton’s gravitational constant, with the energy scale being inversely
proportional to the cosmic time. This concept has been discussed in several
sources, including [35]-[37]. The simplest prototype model of a harmonic
oscillator in such time-dependent NC background was first discussed in [1]
where the time dependent model is exactly solved using Lewis method of
invariant [40, 38, 39]. As the method of invariant as well as the Lewis
invariant is associated with a non-linear differential equation, known as
Ermakov-Pinney equation [41]-[42] in the literature, the equation behaves like
a constraint relation to be obeyed by the parameters those form the
eigenfunction of the Hamiltonian. Thus, every solution set of the EP equation
can provide an explicit form for the eigenfunction of the Hamiltonian, and a
class of exact solutions for the Hamiltonian can be obtained using this
technique. Later, with this time-dependent noncommutativity in place, there
were several works [43]-[45] on the model of time dependent Hamiltonians. In
[43] and [44], using Lewis technique, a class of exact solutions are obtained
for a damped harmonic oscillator both in the absence and presence of an
external magnetic field and placed in the time dependent NC space. In another
recent communication [45], we have used the new change of variables (discussed
above) [2] to study the explicit existence of Berry’s geometric phase [46, 47]
formed in time dependent NC harmonic oscillator systems. Hence, there is a
strong possibility to develop a quantum theory in the background of time
dependent NC space.
In this current study, we examine a time independent harmonic oscillator
situated in a time dependent NC background utilizing coordinate mapping
relations introduced by [2]. We followed the approach taken by [1] where
standard Bopp-shift relations were used to map the original Hamiltonian of the
time-independent harmonic oscillator in NC space into ordinary commutative
variables.

The organization of our work is as follows. In section 2, we consider a two-
dimensional, time-independent quantum harmonic oscillator in time-dependent NC
space associated with a modified version of Bopp-shift relations. We construct
the original Hamiltonian in NC space and then express it in terms of standard
commutative variables, resulting in a Hamiltonian containing a Zeeman term and
a scale-invariant term. In section 3, we use the Lewis-Riesenfeld [40]
approach to solve the resultant time-dependent Hamiltonian, referring to one
of our recent communications [45] that addresses a similar time-dependent
system using the same approach. We begin by presenting the form of the Lewis
invariant associated with the non-linear Ermakov-Pinney (EP) [41]-[42]
equation. Then, we introduce the eigenfunction of the Hamiltonian, which is a
product of the eigenfunction of the Lewis invariant and a time-dependent phase
factor, also known as the Lewis phase in the literature. In section 4, we
expand upon the EP solution set provided in [1] to solve the non-linear EP
equation consistent with the Chiellini integrability condition [48]. In
Section 5, we devise a procedure to calculate the expectation value of the
Hamiltonian with respect to its own eigenstate. Using this expression, we
study the evolution of the energy expectation value of the system with time
for various types of EP solution set, both analytically and graphically. We
also derive the generalized form of the uncertainty equality relations obeyed
by the commutative and NC coordinate operators. Finally, in Section 6, we
summarize our results.

## 2 Model of a harmonic oscillator in modified noncommutative space

We start by looking at a harmonic oscillator in a NC space, incorporating a
recent modification discussed in [2]. The Hamiltonian in the NC space shares a
similar structure with that examined in [1]. The Hamiltonian of the system in
NC space reads,

$H(t)|_{NC}=\dfrac{1}{2\,M}(P_{1}^{2}+P_{2}^{2})+\dfrac{1}{2}\,M\,\omega^{2}(X_{1}^{2}+X_{2}^{2})~{};$
(1)

where $M$ and $\omega$ are the constant mass and angular frequency of the
harmonic oscillator respectively. The commutation relations among the NC
variables which are position and momentum coordinates $(X_{i},P_{i})$, are as
follows (considering, $\hbar=1$)

$\displaystyle[X_{1},X_{2}]=i\theta(t)~{},~{}[P_{1},P_{2}]=i\Omega(t)~{},~{}[X_{1},P_{1}]=i=[X_{2},P_{2}]~{};$
(2)

where $\theta(t)$ and $\Omega(t)$ denote the NC parameters for space and
momentum respectively. The commutation relations among the canonical variables
$(x_{i},p_{i})$ in commutative space are

$[x_{i},p_{j}]=i\delta_{i,j},[x_{i},x_{j}]=0=[p_{i},p_{j}];(i,j=1,2).$ (3)

In order to represent the Hamiltonian [eqn.(1)] in commutative space, we apply
the modified transformation relations between the NC and commutative
coordinates which read [2],

$\displaystyle
X_{1}=x_{1}-\dfrac{\theta(t)}{2}p_{2}+\dfrac{\sqrt{-\theta(t)\Omega(t)}}{2}x_{2}~{},~{}X_{2}=x_{2}+\dfrac{\theta(t)}{2}p_{1}-\dfrac{\sqrt{-\theta(t)\Omega(t)}}{2}x_{1}~{};$
(4) $\displaystyle
P_{1}=p_{1}+\dfrac{\Omega(t)}{2}x_{2}+\dfrac{\sqrt{-\theta(t)\Omega(t)}}{2}p_{2}~{},~{}P_{2}=p_{2}-\dfrac{\Omega(t)}{2}x_{1}-\dfrac{\sqrt{-\theta(t)\Omega(t)}}{2}p_{1}~{};$
(5)

where $\theta(t)\Omega(t)<\,0$.

The Hamiltonian in terms of the commutative variables $(x_{i},p_{i})$ then
takes the form

$H(t)=\dfrac{a(t)}{2}\left({p_{1}}^{2}+{p_{2}}^{2}\right)+\dfrac{b(t)}{2}\left({x_{1}}^{2}+{x_{2}}^{2}\right)+c(t)\left({p_{1}}{x_{2}}-{p_{2}}{x_{1}}\right)+d(t)\left(x_{1}p_{1}+p_{1}x_{1}+x_{2}p_{2}+p_{2}x_{2}\right)\,\,\,;$
(6)

where $a(t),b(t),c(t),d(t)$ are the time dependent coefficients formed as

$\displaystyle
a(t)=\dfrac{1}{M}\left(1-\dfrac{\theta(t)\Omega(t)}{4}\right)+\dfrac{M\,\omega^{2}\theta^{2}(t)}{4}~{},b(t)=M\,\omega^{2}\left(1-\dfrac{\theta(t)\Omega(t)}{4}\right)+\dfrac{\Omega^{2}(t)}{4\,M}~{},$
$\displaystyle
c(t)=\dfrac{\Omega(t)}{2M}+\dfrac{M\omega^{2}\theta(t)}{2}~{},~{}d(t)=\dfrac{\sqrt{-\theta(t)\Omega(t)}}{4}\left[\dfrac{\Omega(t)}{2M}-\dfrac{M\omega^{2}\theta(t)}{2}\right]~{}.$
(7)

Note that while the form of the Hamiltonian [eqn.(6)] and its time-dependent
coefficients $a(t)$, $b(t)$, $c(t)$, and $d(t)$ [eqn.(7)] are similar to those
used in [45] to investigate various explicit forms of geometric phases in NC
space, our current study has different objectives from those in [45].

## 3 Lewis-Riesenfeld approach for solving the model Hamiltonian

We briefly review the Lewis-Riesenfeld approach [40] to determine the
eigenstate of the model Hamiltonian $H(t)$ [Eqn.(6)]. The approach involves
constructing the time-dependent Hermitian invariant operator $I(t)$, also
known as the Ermakov-Lewis invariant, that corresponds to the form of the
Hamiltonian operator for our system. According to this formalism, if we can
determine the eigenstate of the operator $I(t)$,

$I(t)\ket{\phi}=\epsilon\,\ket{\phi}~{};$ (8)

where $\epsilon$ denotes the time independent eigenvalue of $I(t)$
corresponding to its time dependent eigenstate $\ket{\phi}$, then one can also
obtain the time-dependent eigenstate of the operator $H(t)$, denoted by
$\ket{\psi}$, using the following relation provided in [40],

$\ket{\psi}=e^{i\Theta(t)}\ket{\phi}~{}.$ (9)

The time dependent phase factor $\Theta(t)$, commonly referred to as the Lewis
phase, reads

$\dot{\Theta}(t)=\bra{\phi}i\partial_{t}-H(t)\ket{\phi}~{}.$ (10)

###  3.1 The Ermakov-Lewis invariant for our model system and its polar
representation

The initial step in solving the time-dependent model system using the method
of invariants [40] is to obtain the Lewis invariant operator form of $I(t)$.
The invariance of the operator $I(t)$ implies

$\dfrac{dI}{dt}=\partial_{t}{I}+\dfrac{1}{i}[I,H]=0~{}.$ (11)

As demonstrated in [45], the Lewis technique is utilized to solve the
Hamiltonian described in eqn. (6). Thus, for our present study, we adopt the
final expression of $I(t)$ from [45]. The invariant reads

$I(t)=\rho^{2}({p_{1}}^{2}+{p_{2}}^{2})+\left[\dfrac{\left(\dot{\rho}-2\rho\,d\right)^{2}}{a^{2}}+\dfrac{\xi^{2}}{\rho^{2}}\right]({x_{1}}^{2}+{x_{2}}^{2})-\dfrac{\rho}{a}\left(\dot{\rho}-2\rho\,d\right)(x_{1}{p_{1}}+p_{1}{x_{1}}+x_{2}{p_{2}}+p_{2}{x_{2}})~{}$
(12)

where $\rho=\rho(t)$, a time dependent parameter, obeys a non-linear
differential equation given by

$\ddot{\rho}-\dfrac{\dot{a}}{a}\dot{\rho}+\rho\,\left(ab-2\dot{d}-4d^{2}+2\dfrac{\dot{a}}{a}d\right)={\xi^{2}}\dfrac{a^{2}}{\rho^{3}}~{}.$
(13)

In the subsequent discussion, the dimensionless constant of integration,
denoted by $\xi$, will be set to $1$. This non-linear differential equation is
commonly referred to as the Ermakov-Pinney (EP) equation in the literature.
For the sake of convenience in calculations, we present the Lewis invariant
[Eqn.(12)] in the form of polar coordinate variables. 111Note that the polar
coordinate variable $\theta$ should not be confused with the time-dependent NC
parameter $\theta(t)$. To achieve this, we adopt the same procedure as
outlined in our previous communication [43]. The invariant in polar
representation can be expressed as,

$\displaystyle
I(t)=\dfrac{\xi^{2}}{\rho^{2}}r^{2}+\left(\rho{p_{r}}-\dfrac{\dot{\rho}-2\rho\,d}{a}r\right)^{2}+\left({\dfrac{\rho{p_{\theta}}}{r}}\right)^{2}-\left({\dfrac{\rho}{2r}}\right)^{2}~{};$
(14)

where the canonical coordinates in polar representation have the following
form,

$\displaystyle
p_{r}=-i\left({\partial}_{r}+\dfrac{1}{2r}\right)~{}~{},~{}~{}p_{\theta}=-i{\partial_{\theta}}~{}.$
(15)

Here we would like to point out that the time-dependent Hermitian invariant
operator $I(t)$, as obtained, is not unique. Instead, it is possible to get
another alternative form of the invariant operator from the current expression
of $I(t)$ [as given in Eqn.(12)]. This alternative form will prove to be more
advantageous in determining the eigenstate of the Hermitian invariant, and we
will explore this further in the subsequent discussion.

###  3.2 Eigenfunction associated with Ermakov-Lewis invariant

To construct the eigenstate of a Hermitian invariant, we shall now obtain a
more suitable form of the invariant operator using the expression in Eqn.(12).
Once we have the new form of the invariant operator, we can proceed with the
construction of the eigenstate. It is worth noting that this approach has also
been utilized in [1, 45] to solve the invariant system.

####  3.2.1 An alternative expression for the Lewis invariant

Let us define a new form of invariant as,

$I^{{}^{\prime}}=\dfrac{I}{4}-\dfrac{p_{\theta}}{2}.$ (16)

It is very easy to verify that $I^{{}^{\prime}}(t)$ also holds the property of
invariance as $[p_{\theta},H]=0$. Hence, in the discussion to follow, we will
solve the newly defined form of the invariant introduced above to determine
the eigenfunction of the Hamiltonian.

####  3.2.2 Eigenfunction of the alternative form of the invariant

In the first section, we introduced our Hamiltonian $H(t)$, which has the same
structure as the one presented in [45], and it was solved using the Lewis
method of invariant. The detailed discussion of solving the invariant operator
with the appropriate form of ladder operators is provided in [45]. In our
present work, we utilize the obtained expression of the eigenfunction of the
invariant operator [45].

The eigenfunction of the invariant in the polar coordinate system reads [45].

$\displaystyle\phi_{n,m-n}(r,\theta,t)=\dfrac{i^{m}\,\rho^{m-n-1}}{\sqrt{m!n!\pi}}r^{n-m}e^{i(m-n)\theta-\dfrac{a-i\rho(\dot{\rho}-2\rho\,d)}{2a{\rho}^{2}}r^{2}}U\left(-m,1-m+n,\dfrac{r^{2}}{\rho^{2}}\right)~{};$
(17)

where $U\left(-m,1-m+n,\dfrac{r^{2}}{\rho^{2}}\right)$ is known as Tricomi’s
confluent hypergeometric function [49, 50] which can also be expressed in
terms of the associated Laguerre polynomial as

$U\left(-m,1-m+n,\dfrac{r^{2}}{\rho^{2}}\right)=\dfrac{m!}{(-1)^{m}}\,L^{n-m}_{m}\left(\dfrac{r^{2}}{\rho^{2}}\right)~{}.$
(18)

Thus, the eigenfunction of the invariant operator can also be expressed as,

$\displaystyle\phi_{n,m-n}(r,\theta,t)$
$\displaystyle=\dfrac{i^{-m}\sqrt{m!}\,\rho^{m-n-1}}{\sqrt{n!\pi}}r^{n-m}e^{i(m-n)\theta-\dfrac{a-i\rho(\dot{\rho}-2\rho\,d)}{2a{\rho}^{2}}r^{2}}\,L^{n-m}_{m}\left(\dfrac{r^{2}}{\rho^{2}}\right)~{}$
$\displaystyle=Q_{n,m-n}(t)\,R_{n,m-n}(r,t)\,\Phi_{n,m-n}(\theta,t)\,;$ (19)

where the functions $R_{n,m-n}(r,t),\Phi_{n,m-n}(\theta,t)$ and $Q_{n,m-n}(t)$
are given by

$\displaystyle
R_{n,m-n}(r,t)=r^{n-m}e^{-\dfrac{a-i\rho(\dot{\rho}-2\rho\,d)}{2a{\rho}^{2}}r^{2}}\,L^{n-m}_{m}\left(\dfrac{r^{2}}{\rho^{2}}\right)\,,$
$\displaystyle\Phi_{n,m-n}(\theta,t)=e^{i(m-n)\theta}~{},~{}Q_{n,m-n}(t)=\dfrac{i^{-m}\sqrt{m!}\,\rho^{m-n-1}}{\sqrt{n!\pi}}~{}.$
(20)

The eigenfunction $\phi_{n,m-n}(r,\theta,t)$ holds the following
orthonormality condition,

$\int_{0}^{2\pi}d\theta\int_{0}^{\infty}rdr\phi^{*}_{n,m-n}(r,\theta)\phi_{n^{{}^{\prime}},m^{{}^{\prime}}-n^{{}^{\prime}}}(r,\theta)=\delta_{nn^{{}^{\prime}}}\delta_{mm^{{}^{\prime}}}.$
(21)

###  3.3 Lewis phase factor and the eigenfunction of the model Hamiltonian

The Lewis phase factor’s form, originally presented by Lewis et al in [40],
can be derived from the following condition [Eqn.(10)],

$\dot{\Theta}(t)=\bra{\phi}i\partial_{t}-H(t)\ket{\phi}.$ (22)

The form was obtained in [1, 45] and reads,

$\Theta_{n,m-n}(t)=\,m\,\int_{0}^{t}{\left[c(\tau)-\dfrac{a(\tau)}{\rho^{2}(\tau)}\right]d\tau}.$
(23)

To construct the eigenfunction of the Hamiltonian Eqn.(6), we now have all the
ingredients, including Eqn(s).(19, 23). Using equation Eqn.(9), the
eigenfunction of H(t) can be written as follows

$\displaystyle\psi_{n,m-n}(r,\theta,t)$ $\displaystyle=$
$\displaystyle\dfrac{i^{-m}\sqrt{m!}\,\rho^{m-n-1}}{\sqrt{~{}n!\,\pi~{}}}\,exp\left[{i\,m\,\int_{0}^{t}{\left[c(\tau)-\dfrac{a(\tau)}{\rho^{2}(\tau)}\right]d\tau}}\right]$
(24)
$\displaystyle\times\,r^{n-m}e^{i(m-n)\theta-\dfrac{a-i\rho(\dot{\rho}-2\rho\,d)}{2a{\rho}^{2}}r^{2}}L^{n-m}_{m}\left(\dfrac{r^{2}}{\rho^{2}}\right)~{}.$

## 4 Exact analytical solutions of the Ermakov-Pinney equation

In order to obtain the exact analytical solutions of the EP equation, we
essentially follow [1], where the exact analytical solutions of the EP
equation [Eqn.(13)] have been constructed for the case where $d(t)\,=\,0$ by
following the Chiellini integrability condition [48].

Here, we extend their solution set by obtaining the explicit value of the
additional parameter $d(t)$ corresponding to the set of values of the
parameters $a(t)$, $b(t)$, and $\rho(t)$ derived in [1]. it can then be
verified that the parameters $a(t)$, $b(t)$, $\rho(t)$ and $d(t)$ also satisfy
the Chiellini integrability condition.

###  4.1 Exponential Ermakov-Pinney solution

The exponential solution set of EP equation derived in [1] is given by the
following relations,

$\displaystyle a(t)=\sigma e^{-\Gamma{t}}\,\,\,,\,\,\,b(t)=\Delta
e^{\Gamma{t}}\,\,\,,\,\,\rho(t)={\mu}e^{-\Gamma/2}\,\,\,\,\,;$ (25)

where $\sigma,\Delta,\mu$ and $\Gamma$ (positive real number) are the
constants constrained by the following relation

$4\sigma\Delta\mu^{4}-\mu^{4}\Gamma^{2}-4\sigma^{2}=0~{};$ (26)

which appears after the direct substitution of the above solution set
[Eqn.(25)] into the form of the EP equation [Eqn.(13)] at the limit
$d(t)\,=\,0$.

The expressions for $a(t),b(t)$ and $\rho(t)$ are now substituted into the EP
equation (Eqn.[13)]. This results in the emergence of the following
differential equation that is obeyed by the additional parameter $d(t)$,

$\dot{d}+2d^{2}+\Gamma\,d=\dfrac{4\mu^{4}\sigma\Delta-\mu^{4}\Gamma^{2}-4\sigma^{2}}{8\mu^{4}}\equiv\Bbbk\,\text{(constant)}~{}.$
(27)

The solution for $d(t)$ takes the form,

$\displaystyle
d(t)\,=\,\dfrac{1}{4}\left[\sqrt{\Gamma^{2}+8\Bbbk}~{}~{}\dfrac{\complement\,e^{\sqrt{\Gamma^{2}+8\Bbbk}\,\,t}+1}{\complement\,e^{\sqrt{\Gamma^{2}+8\Bbbk}\,\,t}-1}-\Gamma\right]~{}~{};$
(28)

where $\complement$, the integration constant, is set to be greater than unity
to ensure that the obtained solution can no longer diverge at any time $t$. To
elaborate the fact, we derive the critical time $t_{0}$ at which the above
solution would diverge. The condition of divergence for the above solution is
as follows,

$\complement\,e^{\sqrt{\Gamma^{2}+8\Bbbk}\,\,t_{0}}-1=0$ (29)

and the critical time $t_{0}$ found to be

$t_{0}\,=\,-\dfrac{log\,\complement}{\sqrt{\Gamma^{2}+8\Bbbk}}~{}.$ (30)

Hence, by choosing the value of $\complement$ to be greater than $1$, we can
make $t_{0}$ negative. This avoids the divergence, as physical time $t$ cannot
approach $t_{0}$.

#### A special and convenient form of exponential EP solution :

It is worth noting that Eqn.(28) yields various simple forms of the solution
when a suitable value of the constant $\Bbbk$ in Eqn.(27) is chosen. To
understand this better, we consider the simplest example by setting the value
of $\Bbbk$ to zero. This results in the following relation as can be seen from
Eqn.(27),

$\displaystyle\mu^{4}\sigma\Delta-\sigma^{2}=\dfrac{\mu^{4}\Gamma^{2}}{4}~{};$
(31)

which is basically identical to the relation in Eqn.(26). Hence, the parameter
$d(t)$ takes the form

$d(t)|_{\Bbbk=0}=\dfrac{\Gamma}{2\left(\complement\,e^{\Gamma\,t}-1\right)}~{}~{};~{}~{}\complement\,>1~{}.$
(32)

Note that $d(t)$ becomes zero in the limit $\complement\rightarrow\infty$.

####  4.1.1 Consistency with the Chilleni integrability condition

In this subsection, we aim to demonstrate that these solution sets discussed
earlier also satisfy the Chilleni integrability condition [48], which was used
in [1] to derive the explicit expressions for the first three EP parameters,
namely $a(t)$, $b(t)$, and $d(t)$. To accomplish this, we shall first write
down the Chilleni condition below.

Defining certain components of the EP equation (Eqn. 13) as follows

$\displaystyle\dot{\rho}=\eta~{},~{}g(\rho)=-\dfrac{\dot{a}}{a}~{},~{}h(\rho)=\rho\left(ab-2\dot{d}-4\,d^{2}+2\dfrac{\dot{a}}{a}d\right)-\dfrac{a^{2}}{\rho^{3}}~{};$
(33)

enables to transform Eqn.(13) into the following first order differential
equation

$\eta\dfrac{d\eta}{d\rho}+\eta\,g(\rho)+h(\rho)=0~{}.$ (34)

Now the Chilleni integrability condition states that if

$\displaystyle\dfrac{d}{d\rho}\left(\dfrac{h(\rho)}{g(\rho)}\right)=q\,g(\rho)~{};~{}(q=\text{constant})$
(35)

then the unknown parameter $\eta$ has the following solution as [48]

$\displaystyle\eta=\lambda_{q}\dfrac{h(\rho)}{g(\rho)}~{}~{}~{}\text{with}~{}~{}~{}\lambda_{q}=\dfrac{-1\pm\sqrt{1-4\,q}}{2\,q}~{}.$
(36)

Substituting the expressions for exponential EP solution set
$a(t),b(t),\rho(t)$ and $d(t)$ together with the constraint relation
[Eqn.(27)] in Eqn.(33), the explicit form of $\eta(\rho),g(\rho)$ and
$h(\rho)$ are calculated to be

$\displaystyle\eta(\rho)=-\dfrac{\Gamma}{2}\rho~{}~{},~{}~{}g(\rho)=\Gamma~{}~{},~{}~{}h(\rho)=\dfrac{\Gamma^{2}}{4}\rho~{}.$
(37)

With the above set of values in our hand, the Chilleni integrability condition
stated in Eqn(s).(35, 36) can be easily shown to be satisfied. The
corresponding values of the constants $q$ and $\lambda_{q}$, for this solution
set, are computed to be

$\displaystyle q=\dfrac{1}{4}~{}~{},~{}~{}\lambda_{q}=-2~{};$ (38)

which also obey the relation among $q$ and $\lambda_{q}$.

###  4.2 Rational Ermakov-Pinney solution

The rational solution set of EP equation derived in [1] is given by the
following relations,

$\displaystyle
a(t)=\dfrac{\sigma\,\left(1+\dfrac{2}{k}\right)^{\,(k+2)/k}}{(\Gamma{t}+\chi)^{\,(k+2)/k}}~{},~{}b(t)=\dfrac{\Delta\,\left(\dfrac{k}{k+2}\right)^{(2-k)/k}}{(\Gamma{t}+\chi)^{\,(k-2)/k}}~{},~{}\rho(t)=\dfrac{\mu\left(1+\dfrac{2}{k}\right)^{1/k}}{(\Gamma{t}+\chi)^{1/k}}~{};$
(39)

where $k\,\,\in\,\,N$ and $\sigma,\Delta,\mu,\chi$ and $\Gamma$ (positive real
number) are the constants constrained by the following relation

$\left(k+2\right)^{2}\left(\sigma\Delta\mu^{4}-\sigma^{2}\right)-\mu^{4}\Gamma^{2}=0~{};$
(40)

which appears after the direct substitution of the above solution set
[Eqn.(39)] in the reduced form of the EP equation [Eqn.(13)] with $d=\,0$.
Now we substitute the expression of $a(t),b(t)\,$and$\,\rho(t)$ in the EP
equation [Eqn.(13)]. This leads to the following differential equation for the
additional parameter $d(t)$,

$\displaystyle\dfrac{1}{k^{2}\left(\Gamma{t}+\chi\right)^{2}}\left[\left(k+2\right)^{2}\left(\sigma\Delta\mu-\dfrac{\sigma^{2}}{\mu^{3}}\right)-\mu\Gamma^{2}\right]=2\mu\left[\dot{d}(t)+2d^{2}(t)+d(t)\dfrac{\Gamma(k+2)}{k(\Gamma\,t+\chi)}\right]~{}.$
(41)

It can be seen that the time dependent factors in the above equation would
cancel if the unknown parameter $d(t)$ has the following form

$d(t)=\dfrac{\delta}{\left(\Gamma\,t+\chi\right)}~{};$ (42)

where $\delta$ is a positive, real constant. With the above form of $d(t)$,
Eqn.(41) now reduces to a constraint relation of the form,

$4k^{2}\mu^{4}\delta\left(\delta+\dfrac{\Gamma}{k}\right)=\left(k+2\right)^{2}\left(\sigma\Delta\mu^{4}-\sigma^{2}\right)-\mu^{4}\Gamma^{2}~{}.$
(43)

Hence, the EP solution in terms of rational functions are given by Eqn(s).(39,
42) with the constraint relation Eqn.(43). It should be noted that while the
parameters $a(t)$, $b(t)$, and $\rho(t)$ take on various special forms due to
different values of $k$, the parameter $d(t)$ remains entirely independent of
$k$.

####  4.2.1 Consistency with the Chilleni integrability condition

In this subsection, we show that the rational solution sets satisfy the
Chilleni integrability condition discussed in subsection $4.1.1$. To do so, we
substitute the expressions for the rational EP solution set, namely,
$a(t),b(t),\rho(t),$ and $d(t),$ along with the constraint relation
[Eqn.(43)], into Eqn.(33). By doing this, we obtain explicit expressions for
$\eta(\rho),$ $g(\rho),$ and $h(\rho)$,

$\displaystyle\eta(\rho)=-\dfrac{\rho^{k+1}\Gamma}{\mu^{k}(k+2)}~{}~{};~{}~{}g(\rho)=\Gamma\dfrac{\rho^{k}}{\mu^{k}}~{}~{};~{}~{}h(\rho)=\dfrac{\rho^{2k+1}\Gamma^{2}}{\mu^{2k}\left(k+2\right)^{2}}~{}.$
(44)

With the above set of values in our hand, the Chilleni integrability condition
stated in Eqn(s).(35, 36) can easily be verified and the corresponding values
of the constants $q$ and $\lambda_{q}$ are given by

$\displaystyle
q=\dfrac{k+1}{\left(k+2\right)^{2}}~{}~{},~{}~{}\lambda_{q}=-\left(k+2\right)~{};$
(45)

which also obey the relation $\lambda_{q}=(-1-\sqrt{1-4\,q})\,/2\,q$ among $q$
and $\lambda$.

## 5 Expectation values and uncertainty relations

We now aim to obtain the expression for energy expectation value, and the
uncertainty relations among the coordinate and momentum operators.

The expression for $\langle H\rangle$, arising from Eqn[6], is given by

$\langle
H\rangle=\dfrac{a(t)}{2}\left(\langle{p_{1}}^{2}\rangle+\langle{p_{2}}^{2}\rangle\right)+\dfrac{b(t)}{2}\left(\langle{x_{1}}^{2}\rangle+\langle{x_{2}}^{2}\rangle\right)+c(t)\left(\langle{p_{1}}{x_{2}}-{p_{2}}{x_{1}}\rangle\right)+d(t)\left(\langle{x_{1}}{p_{1}}+{p_{1}}{x_{1}}\rangle+\langle{x_{2}}{p_{2}}+{p_{2}}{x_{2}}\rangle\right)\,\,\,.$
(46)

Note that the eigenstates of the Hamiltonian $H(t)$ are denoted by
$|n,l\rangle_{H}$.

###  5.1 Expectation values of coordinate operators, bilinear products, and
the operators raised to the second power

To begin with, we present the expectation values of the configuration space
operators $x_{1}$ and $x_{2}$, as well as their respective squared forms,
$x_{1}^{2}$ and $x_{2}^{2}$. It is noteworthy that the computation of these
quantities, when raised to any arbitrary finite power, has been previously
elaborated in [43] where we investigated a damped harmonic oscillator in a
time-dependent background of noncommutativity. The expectation values can be
written as follows [1, 45],

${}_{H}\langle n,m-n|\,x_{i}\,|n,m-n\rangle_{H}\,$ $\displaystyle=\,0~{},$
${}_{H}\langle n,m-n|\,x_{i}^{2}\,|n,m-n\rangle_{H}\,$
$\displaystyle=\,\dfrac{(m+n+1)}{2}\,\rho^{2}~{};$ (47)

where $i=1,2$. Our results, obtained using a generalized version of the Bopp-
shift relations, are consistent with those from [1, 43], where a time-
dependent NC framework was developed using standard Bopp-Shift relations. It
happens due to the fact that, in our generalized eigenfunction (Eqn.[24]) ,
the extra parameter $d(t)$ generated due to the modification included in the
coordinate mapping relations, exists as a phase. As a result, the factor
$d(t)$ does not contribute since it is eliminated during the calculation of
expectation value.

We will now calculate the expectation values of the momentum operators,
$p_{1}$ and $p_{2}$, as well as their respective squared forms, $p_{1}^{2}$
and $p_{2}^{2}$. These operators can be expressed in polar form as follows,

$\displaystyle
p_{1}\,=\,\left[-i\,cos\,\theta\,\partial_{r}+\dfrac{i}{r}sin\theta\,\partial_{\theta}\right]~{},~{}p_{2}\,=\,\left[-i\,sin\,\theta\,\partial_{r}-\dfrac{i}{r}cos\theta\,\partial_{\theta}\right]~{}.$
(48)

Since the detailed calculation reveals that both $\braket{p_{1}}$ and
$\braket{p_{2}}$ yield trivial zero results, we just present those in the
following manner

${}_{H}\langle n,m-n|\,p_{i}\,|n,m-n\rangle_{H}\,=\,0\,;~{}~{}i\,=\,[1,2]\,,$
(49)

and then proceed to calculate $\braket{\,p_{1}^{2}\,}$ and
$\braket{\,p_{2}^{2}\,}$. We mention some of the intermediate calculational
steps in getting the final result. The expectation value of $p_{1}^{2}$ reads

${}_{H}\langle n,m-n|\,p_{1}^{2}\,|n,m-n\rangle_{H}$ $\displaystyle=$
$\displaystyle\int rdrd\theta~{}_{H}\langle n,m-n|r,\theta\rangle\langle
r,\theta|\left(-i\,cos\,\theta\,\partial_{r}+\dfrac{i}{r}sin\theta\,\partial_{\theta}\right)^{2}|n,m-n\rangle_{H}$
(50) $\displaystyle=$ $\displaystyle\int rdrd\theta~{}\langle
n,m-n|r,\theta\rangle\langle
r,\theta|\left(-i\,cos\,\theta\,\partial_{r}+\dfrac{i}{r}sin\theta\,\partial_{\theta}\right)^{2}|n,m-n\rangle$
$\displaystyle=$
$\displaystyle\int~{}\phi_{n,m-n}^{*}(r,\theta,t)\left(-i\,cos\,\theta\,\partial_{r}+\dfrac{i}{r}sin\theta\,\partial_{\theta}\right)^{2}\phi_{n,m-n}(r,\theta,t)\,rdrd\theta~{};$
$\displaystyle=$ $\displaystyle\,I_{1}+I_{2}+I_{3}+I_{4}+I_{5};$

where we have used the relations
$|n,m-n\rangle_{H}~{}=~{}e^{i\Theta_{n,m-n}}|n,m-n\rangle$ where
$|n,m-n\rangle_{H}$ and $|n,m-n\rangle$ denote the eigenstates of the
Hamiltonian $H(t)$ and Lewis invariant $I(t)$ respectively. We have also
employed the relation $\langle r,\theta|n,m-n\rangle$ =
$\phi_{n,m-n}(r,\theta)$, with $\phi$ presenting the eigenfunction of $I(t)$.
We now calculate the terms $I_{1},I_{2},I_{3},I_{4}$ and $I_{5}$.

${}_{H}\langle
n,m-n|\,p_{1}^{2}\,|n,m-n\rangle_{H}=\,I_{1}+I_{2}+I_{3}+I_{4}+I_{5};$ (51)
$\displaystyle I_{1}=$
$\displaystyle-\int\,\phi_{n,m-n}^{*}(r,\theta,t)\,cos^{2}\theta\,\partial_{r}^{2}\,\left[\phi_{n,m-n}(r,\theta,t)\right]\,r\,dr\,d\theta$
$\displaystyle=-Q^{*}_{n,m-n}(t)\,Q_{n,m-n}(t)\,\int_{0}^{\infty}\,R_{n,m-n}^{*}(r,t)\,\left[\partial_{r}^{2}\,R_{n,m-n}(r,t)\right]\,r\,dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos^{2}\theta\,\Phi_{n,m-n}(\theta,t)\,d\theta~{},$
(52) $\displaystyle I_{2}$
$\displaystyle=-2\int\,\phi_{n,m-n}^{*}(r,\theta,t)\,\left(\dfrac{sin\,\theta\,cos\theta}{r^{2}}\right)\,\partial_{\theta}\,\left[\phi_{n,m-n}(r,\theta,t)\right]\,r\,dr\,d\theta$
$\displaystyle=-2\,Q^{*}_{n,m-n}(t)\,Q_{n,m-n}(t)\,\int_{0}^{\infty}\,\dfrac{R_{n,m-n}^{*}(r,t)\,\,R_{n,m-n}(r,t)}{r}\,dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos\theta\,sin\theta\,\partial_{\theta}\left[\Phi_{n,m-n}(\theta,t)\right]\,d\theta~{},$
(53) $\displaystyle I_{3}$
$\displaystyle=2\int\,\phi_{n,m-n}^{*}(r,\theta,t)\,\left(\dfrac{sin\,\theta\,cos\theta}{r}\right)\,\partial_{r}\partial_{\theta}\,\left[\phi_{n,m-n}(r,\theta,t)\right]\,r\,dr\,d\theta$
$\displaystyle=2\,Q^{*}_{n,m-n}(t)\,Q_{n,m-n}(t)\,\int_{0}^{\infty}\,R_{n,m-n}^{*}(r,t)\,\partial_{r}\,\left[R_{n,m-n}(r,t)\,\right]dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos\theta\,sin\theta\,\partial_{\theta}\left[\Phi_{n,m-n}(\theta,t)\right]\,d\theta~{},$
(54) $\displaystyle I_{4}$
$\displaystyle=-\int\,\phi_{n,m-n}^{*}(r,\theta,t)\,\left(\dfrac{sin^{2}\theta}{r}\right)\,\partial_{r}\,\left[\phi_{n,m-n}(r,\theta,t)\right]\,r\,dr\,d\theta$
$\displaystyle=-Q^{*}_{n,m-n}(t)\,Q_{n,m-n}(t)\,\int_{0}^{\infty}\,R_{n,m-n}^{*}(r,t)\,\left[\partial_{r}\,R_{n,m-n}(r,t)\right]\,dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,sin^{2}\theta\,\Phi_{n,m-n}(\theta,t)\,d\theta~{},$
(55) $\displaystyle I_{5}$
$\displaystyle=-\int\,\phi_{n,m-n}^{*}(r,\theta,t)\,\left(\dfrac{sin^{2}\theta}{r^{2}}\right)\,\partial_{\theta}^{2}\,\left[\phi_{n,m-n}(r,\theta,t)\right]\,r\,dr\,d\theta$
$\displaystyle=-Q^{*}_{n,m-n}(t)\,Q_{n,m-n}(t)\,\int_{0}^{\infty}\,\dfrac{R_{n,m-n}^{*}(r,t)\,R_{n,m-n}(r,t)}{r}\,dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,sin^{2}\theta\,\partial_{\theta}^{2}\,\left[\Phi_{n,m-n}(\theta,t)\right]\,d\theta~{}.$
(56)

Upon observing the azimuthal portions of the above integrals in Eqn(s).(52,
53, 54, 55, 56) , it becomes clear that the terms $I_{2}$ and $I_{3}$ vanish.
This can be seen as

$\displaystyle\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos\theta\,sin\theta\,\partial_{\theta}\left[\Phi_{n,m-n}(\theta,t)\right]\,d\theta$
$\displaystyle=\,i\,(m-n)\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos\theta\,sin\theta\,\Phi_{n,m-n}(\theta,t)\,d\theta$
$\displaystyle=0~{}.$ (57)

As a result, we have $I_{2}\,=\,0\,=I_{3}$. To proceed further, we insert the
expressions for $R$, $\Phi$, and $Q$ from Eqn. [20] into the remaining
variables $I_{1}$, $I_{4}$, and $I_{5}$, and subsequently introduce the
variable $z$ to parametrize $\dfrac{r^{2}}{\rho^{2}}$. In order to simplify
the integrations, we utilize numerous valuable and specific relationships such
as the derivative form, an identity relation, and the orthonormality condition
of the associated Laguerre polynomial. The derivative of the associated
Laguerre polynomial reads [49],

$\displaystyle\partial_{z}\left[L^{n-m}_{m}\,\left(z\right)\right]$
$\displaystyle=-\,L^{n-m+1}_{m-1}\,\left(z\right)~{};~{}(m\,\geq\,1)$
$\displaystyle=\,0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{};(m=0)~{}.$

The orthonormality condition and the identity relation for the associated
Laguerre polynomial reads [49]

$\displaystyle\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{n}(z)\,L^{n-m}_{m}(z)\,=\,\dfrac{\Gamma\,(2n-m+1)}{n!}\,\delta_{mn}~{},$
$\displaystyle L^{n-m}_{m}(z)\,=\,L^{n-m+1}_{m}(z)-L^{n-m+1}_{m-1}(z)~{}.$
(59)

Using the above relations (Eqn.[59]) we can calculate the expressions
$I_{1},I_{4}$, and $I_{5}$ which read

$\displaystyle I_{1}$
$\displaystyle=-\dfrac{m!\,(n-m)(n-m-1)}{2\,n!\,\rho^{2}}\,\int^{\infty}_{0}\,z^{n-m-1}\,e^{-z}\,L^{n-m}_{m}(z)L^{n-m}_{m}(z)\,dz\,+\,(2n+2m+1)\,\dfrac{f}{2\rho}-\dfrac{f^{2}}{2}(m+n+1)$
$\displaystyle+(2n-2m+1)\dfrac{m!}{\rho^{2}\,n!}\,\int_{0}^{\infty}z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)L^{n-m+1}_{m-1}(z)\,dz-\dfrac{2\,m!}{n!\rho^{2}}\,\int_{0}^{\infty}\,z^{n-m+1}\,e^{-z}\,L^{n-m}_{m}(z)L^{n-m+2}_{m-2}(z)\,dz,$
(60)

$\displaystyle
I_{4}=-\dfrac{m!\,(n-m)}{2\,n!\,\rho^{2}}\,\int^{\infty}_{0}\,z^{n-m-1}\,e^{-z}\,L^{n-m}_{m}(z)L^{n-m}_{m}(z)\,dz\,+\,\dfrac{f}{2\rho}+\dfrac{m!}{\rho^{2}\,n!}\,\int_{0}^{\infty}z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)L^{n-m+1}_{m-1}(z)\,dz~{},$
(61) $\displaystyle
I_{5}=\dfrac{m!\,(m-n)^{2}}{2\,n!\,\rho^{2}}\,\int^{\infty}_{0}\,z^{n-m-1}\,e^{-z}\,L^{n-m}_{m}(z)L^{n-m}_{m}(z)\,dz~{}$
(62)

where $f=\dfrac{a-i\rho(\dot{\rho}-2\rho\,d)}{a\rho}$. Summing them up,

${}_{H}\langle n,m-n|\,p_{1}^{2}\,|n,m-n\rangle_{H}$
$\displaystyle=\,\dfrac{2(n-m+1)\,m!}{n!\,\rho^{2}}\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+1}_{m-1}(z)\,dz$
$\displaystyle-\,\dfrac{2\,m!}{n!\,\rho^{2}}\,\int_{0}^{\infty}\,z^{n-m+1}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+2}_{m-2}(z)\,dz$
$\displaystyle+\dfrac{(n+m+1)}{2}\,\left[\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right]~{}.$
(63)

Now we use the following identity (derived in the Appendix A)

$(n-m+1)\,\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+1}_{m-1}(z)\,dz=\,\int_{0}^{\infty}\,z^{n-m+1}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+2}_{m-2}(z)\,dz~{}.$
(64)

Using the above identity, Eqn.(63) reduces to the form

${}_{H}\langle
n,m-n|\,p_{1}^{2}\,|n,m-n\rangle_{H}\,=\,\dfrac{(n+m+1)}{2}\,\left[\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right]~{};$
(65)

which is a real and positive quantity. The presence of the parameter $d(t)$ in
the above expression enables us to visualize the effect of employing the
generalized coordinate mapping relation between NC space and commutative
space. When the parameter $d(t)$ vanishes, our result reduces to that
previously obtained in [1]. Furthermore, in our system, the appearance of
$d(t)$ is solely due to presence of NC parameters, allowing us to identify its
presence as a pure NC effect in the above expression.

We can follow a similar procedure to obtain the expectation value of
$p_{2}^{2}$., which yields the same result as $\braket{p_{1}^{2}}$. For this
we have,

${}_{H}\langle
n,m-n|\,p_{2}^{2}\,|n,m-n\rangle_{H}\,=\,\dfrac{(n+m+1)}{2}\,\left[\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right]~{}.$
(66)

Eqn.(s)(65, 65) can be expressed in the following manner,

${}_{H}\langle
n,m-n|\,p_{i}^{2}\,|n,m-n\rangle_{H}\,=\,\dfrac{(n+m+1)}{2}\,\left[\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right]\,;~{}~{}i\,=\,[1,2]\,.$
(67)

Our next goal is to determine the expectation value of
$\left(x_{1}\,p_{1}+p_{1}\,x_{1}\right)$. To accomplish this, we must first
obtain the polar form of the said quantity. This reads

$\displaystyle\left(x_{1}\,p_{1}+p_{1}\,x_{1}\right)=2\,x_{1}\,p_{1}\,-i$
$\displaystyle=\,-2i\,\left(\,r\,cos^{2}\theta\,\partial_{r}-cos\theta\,sin\theta\,\partial_{\theta}\,\right)-i$
$\displaystyle~{}.$ (68)

We shall now employ the same methodology utilized in computing
$\braket{p_{i}^{2}}$. This now gives,

${}_{H}\langle n,m-n|\left(x_{1}\,p_{1}+p_{1}\,x_{1}\right)|n,m-n\rangle_{H}$
$\displaystyle=\int rdrd\theta~{}_{H}\langle n,m-n|r,\theta\rangle\langle
r,\theta|\left[\,-2i\,\left(\,r\,cos^{2}\theta\,\partial_{r}-cos\theta\,sin\theta\,\partial_{\theta}\,\right)-i\right]|n,m-n\rangle_{H}$
$\displaystyle=\int rdrd\theta~{}\langle n,m-n|r,\theta\rangle\langle
r,\theta|\,\left[\,-2i\,\left(\,r\,cos^{2}\theta\,\partial_{r}-cos\theta\,sin\theta\,\partial_{\theta}\,\right)-i\right]|n,m-n\rangle$
$\displaystyle=-i-2i\int~{}\phi_{n,m-n}^{*}(r,\theta,t)\,\left(\,r\,cos^{2}\theta\,\partial_{r}-cos\theta\,sin\theta\,\partial_{\theta}\,\right)\phi_{n,m-n}(r,\theta,t)\,rdrd\theta$
$\displaystyle=\,-i+I_{1}^{{}^{\prime}}+I_{2}^{{}^{\prime}}~{};$ (69)

where

$\displaystyle
I_{1}^{{}^{\prime}}=-2i\,Q^{*}_{n,m-n}\,Q_{n,m-n}\,\int_{0}^{\infty}\,R_{n,m-n}^{*}(r,t)\,r^{2}\,\left[\partial_{r}\,R_{n,m-n}(r,t)\right]\,dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos^{2}\theta\,\Phi_{n,m-n}(\theta,t)\,d\theta$
(70) $\displaystyle
I_{2}^{{}^{\prime}}=2i\,Q^{*}_{n,m-n}(t)\,Q_{n,m-n}(t)\,\int_{0}^{\infty}\,R_{n,m-n}^{*}(r,t)\,R_{n,m-n}(r,t)\,r\,dr\,\int_{0}^{2\Pi}\,\Phi_{n,m-n}^{*}(\theta,t)\,cos\theta\,sin\theta\,\partial_{\theta}\left[\Phi_{n,m-n}(\theta,t)\right]\,d\theta~{}.$
(71)

Referring back to Eqn.(57), we can conclude that

$I_{2}^{{}^{\prime}}=0~{}.$ (72)

We proceed to evaluate the integral $I_{1}^{\prime}$ using Eqn. (59), which
leads us to the final result of $\langle\,x_{1}p_{1}+p_{1}x_{1}\rangle$ as,

${}_{H}\langle
n,m-n|\left(x_{1}\,p_{1}+p_{1}\,x_{1}\right)|n,m-n\rangle_{H}=(m+n+1)\,\dfrac{\rho\,\left(\dot{\rho}-2\rho\,d\right)}{a}~{}.$
(73)

A similar calculation reveals that the quantity $\langle
x_{2}p_{2}+p_{2}x_{2}\rangle$ also yields the result that we obtained in
Eqn.(73). Therefore, we can represent it separately below,

${}_{H}\langle
n,m-n|\left(x_{2}\,p_{2}+p_{2}\,x_{2}\right)|n,m-n\rangle_{H}=(m+n+1)\,\dfrac{\rho\,\left(\dot{\rho}-2\rho\,d\right)}{a}~{}.$
(74)

or alternatively, present both of them in the following manner

${}_{H}\langle
n,m-n|\left(x_{i}\,p_{i}+p_{i}\,x_{i}\right)|n,m-n\rangle_{H}=(m+n+1)\,\dfrac{\rho\,\left(\dot{\rho}-2\rho\,d\right)}{a}~{};\,i=1,2\,.$
(75)

The last quantity that remains to be calculated is $\braket{H(t)}$ is
$\braket{x_{2}\,p_{1}-p_{2}\,x_{1}}$, which in terms of polar coordinates
reads

$\left(x_{2}\,p_{1}-p_{2}\,x_{1}\right)\,=\,i\partial_{\theta}~{}.$ (76)

The corresponding expectation value which can be derived much more simply than
those we elaborated for the previous quantities, is mentioned below as
follows,

${}_{H}\langle
n,m-n|\left(x_{2}\,p_{1}-p_{2}\,x_{1}\right)|n,m-n\rangle_{H}=(n-m)~{};$ (77)

while each part of the term $\braket{x_{2}\,p_{1}-p_{2}\,x_{1}}$ individually
reads

${}_{H}\langle
n,m-n|\left(x_{1}\,p_{2}\right)|n,m-n\rangle_{H}=\dfrac{(m-n)}{2}~{};~{}_{H}\langle
n,m-n|\left(x_{2}\,p_{1}\right)|n,m-n\rangle_{H}=\dfrac{(n-m)}{2}~{}.$ (78)

Now that we have all the necessary results to compute $\braket{H(t)}$, we
proceed to calculate the expectation values of two more bilinear operators,
namely, $x_{1}x_{2}$ and $p_{1}p_{2}$. The expectation values of these
operators are required to compute the uncertainty relations for the NC
variables which we shall discuss later.

$\displaystyle x_{1}x_{2}\,=$ $\displaystyle\,r^{2}\,sin\theta\,cos\theta~{},$
$\displaystyle
p_{1}\,p_{2}\,=\,-sin\theta\,cos\theta\,\partial_{r}^{2}+\dfrac{cos\,2\theta}{r^{2}}\,\partial_{\theta}-$
$\displaystyle\dfrac{cos\,2\theta}{r}\partial_{r}\partial_{\theta}+\dfrac{sin\theta\,\cos\theta}{r}\partial_{r}+\dfrac{sin\theta\,cos\theta}{r^{2}}\partial_{\theta}^{2}~{}.$
(79)

Carefully observing the azimuthal parts of these quantities, with the
experiences from the previous calculations in mind, we can easily recognize
that neither of these quantities make any contribution to their expectation
value in the eigenstate of $H(t)$. Hence, we mention the following outcomes,

$\displaystyle\braket{x_{1}\,x_{2}}=0~{},~{}\braket{p_{1}\,p_{2}}=0~{}.$ (80)

###  5.2 Energy expectation values: Explicit form and their graphical
investigation

This section is dedicated to explore the time evolution of the energy
expectation value of the system, utilizing both analytical and graphical
representations. In the previous section, we have already performed the
necessary calculations to compute the expectation value of the Hamiltonian
$H(t)$ in its own eigenstate $\psi_{n,m-n}(r,\theta)$. Hence, the expectation
value of energy $\braket{E_{n,m-n}(t)}$ with respect to energy eigenstate
$\psi_{n,m-n}(r,\theta,t)$ can be derived from Eqn.(46) as

$\displaystyle\braket{E_{n,m-n}(t)}$
$\displaystyle=\dfrac{1}{2}\,(n+m+1)\left[b(t)\rho^{2}(t)+\dfrac{a(t)}{\rho^{2}(t)}+\dfrac{\dot{\rho}^{2}(t)-4\rho^{2}(t)\,d^{2}(t)}{a(t)}\right]+c(t)\,(n-m)~{};$
(81)

which in the limit $d(t)\rightarrow\,0,$ gives the expectation value obtained
in [43].

In a previous section discussing the solution of the EP equation, we derived
the explicit functional form of the Hamiltonian parameter as well as the EP
parameter $d(t)$ based on the previously calculated EP parameters $a(t),b(t)$,
and $\rho(t)$, which were presented in [1]. Now we aim to compute the explicit
form of the energy expression in Eqn.(81) by substituting these various
solution sets of the EP equation. Although the Hamiltonian parameter $c(t)$ in
the energy expression above is not considered an EP parameter, it can still be
explicitly calculated, as demonstrated in [43]. By substituting the explicit
form of the parameters, appearing in the Hamiltonian, as well as the EP
parameters $a(t)$, $b(t)$, and $d(t)$, into their respective relations shown
in Eqn.(7), one can obtain the explicit expressions of the NC parameters
$\theta(t)\Omega(t)$, which in turn reveal the explicit form of the parameter
$c(t)$ for constant value of the mass $M$ and the angular frequency
$\omega_{0}$. To simplify things, we proceed with Eqn.(81) $n=m$. This gives

$\displaystyle\braket{E_{m,0}(t)}$
$\displaystyle=\dfrac{1}{2}\,(2\,m+1)\left[b(t)\rho^{2}(t)+\dfrac{a(t)}{\rho^{2}(t)}+\dfrac{\dot{\rho}^{2}(t)-4\rho^{2}(t)\,d^{2}(t)}{a(t)}\right]~{}.$
(82)

####  5.2.1 Energy expectation values : exponential EP solutions

Considering the exponentially varying solution expressed in Eqn.(25) and
Eqn.(32) subject to the constraint relation in Eqn.(31), we can derive the
expression for the energy expectation value, with $n=m$, as follows

$\displaystyle\braket{\,H(t)\,}|_{m=n}^{exp}\,=\,(2m+1)\left[\Delta\,\mu^{2}-\dfrac{\Gamma^{2}\,\mu^{2}}{2\,\sigma\,\left(\complement\,e^{\Gamma\,t}-1\right)^{2}}\right]~{};(\,\complement\,>\,1)~{}.$
(83)

By employing the limit $\complement\rightarrow\infty$, the above expression
reduces to a constant value, which represents the energy associated with the
exponential EP solution in [43], while considering $n=m$. Both the analytic
expression (Eqn. [83]) and its graphical representation (see Fig. [1]) reveal
that for real, positive values of the constants, the energy increases with
time and eventually reaches a finite value at large time.

Figure 1: Plot of the variation of expectation value of energy, (scaled by
$\frac{1}{\omega_{0}}$ ($\frac{\langle E_{m,0}\rangle}{\omega_{0}}$) in order
to make it dimensionless), as we vary $\Gamma$t. The expectation value of
energy $\langle E\rangle$ is calculated for exponentially varying Hamiltonian
parameters as well as EP parameters when $\langle A\rangle$ the system is
mapped by the generalized version of the Bopp-shift relations and the
parameters are considered as $m,n,\mu,\Gamma=1$ and
$\Delta=\dfrac{5}{4},\,\complement=2$; $\langle B\rangle$ the system is mapped
by the standard Bopp-shift relations and the parameters are considered as
$m,\mu,\omega_{0},\sigma,\Gamma=1$, $\Delta=\dfrac{5}{4}$ and
$\complement\,\rightarrow\,\infty$. While for $\langle A\rangle$ the energy
exhibits an initial increase, followed by a saturation, for $\langle B\rangle$
the energy value always remains constant.

####  5.2.2 Energy expectation values : rational EP solutions

Considering the rationally varying solution expressed in Eqn.(39) and Eqn.(42)
subject to the constraint relation in Eqn.(43), we can derive the expression
for the energy expectation value, with $n=m$, as follows

$\displaystyle\braket{\,H(t)\,}|_{m=n}^{rat}\,=\,\dfrac{(2m+1)}{\left(\Gamma\,t+\chi\right)}\left[\dfrac{(k+2)^{2}\,\left(\sigma^{2}+\Delta\mu^{4}\sigma\right)+\Gamma^{2}\mu^{4}-4\mu^{4}\delta^{2}k^{2}}{2\,(k+2)\,\mu^{2}\,k\,\sigma\,}\right]~{}.$
(84)

In the limit $\delta\rightarrow\,0$ the above expression represents the energy
that corresponds to the rational EP solution in [43], when $n=m$ is taken into
consideration. Although the energy decays rationally over time in both cases,
the coefficients of the decay are altered due to the removal of the modified
term from the coordinate mapping relation (Eqn.[5]). This is also reflected
from the graphical representation (Fig.[2]) of the energy expression in
Eqn.(84) .

Figure 2: Plot of the variation of expectation value of energy, (scaled by
$\frac{1}{\omega_{0}}$ ($\frac{\langle E_{m,0}\rangle}{\omega_{0}}$) in order
to make it dimensionless), as we vary $\Gamma$t. The expectation value of
energy $\langle E\rangle$ is calculated for rationally varying Hamiltonian
parameters as well as EP parameters when $\langle A\rangle$ the system is
mapped by the generalized version of the Bopp-shift relations and the
parameters are considered as
$m,n,k,\mu,\Gamma,\omega_{0},\delta,\sigma,\chi=1$ and $\Delta=2$; $\langle
B\rangle$ the system is mapped by the standard Bopp-shift relations and the
parameters are considered as $m,k,\mu,\Gamma,\sigma,\chi=1$ and
$\Delta=\dfrac{10}{9}$. In both cases, the energy, containing different values
of constant coefficients, decays rationally with respect to time and
approaches zero for very large values of time.

###  5.3 Modified uncertainty equality relations for commutative and
noncommutative variables

In this section, our objective is to find uncertainty relations between the
operators.

#### Equal uncertainty products for commutative operators :

The variances for the variables $x_{1}$ and $x_{2}$ are obtained from Eqn.(47)
as

$\displaystyle\Delta\,x_{1}=\sqrt{\braket{x_{1}^{2}}\,-\,\braket{x_{1}}^{2}}\,=\,\dfrac{\sqrt{m+n+1}}{\sqrt{2}}\,\rho~{},~{}~{}\Delta\,x_{2}=\sqrt{\braket{x_{2}^{2}}\,-\,\braket{x_{2}}^{2}}\,=\,\dfrac{\sqrt{m+n+1}}{\sqrt{2}}\,\rho~{};$
(85)

and the same for the variables $p_{1}$ and $p_{2}$ are deduced from
Eqn(s).(49, 67) as

$\displaystyle\Delta\,p_{1}=\sqrt{\braket{p_{1}^{2}}\,-\,\braket{p_{1}}^{2}}\,=\dfrac{\sqrt{m+n+1}}{a\,\rho}\sqrt{\dfrac{a^{2}+\rho^{2}\left(\dot{\rho}-2\rho\,d\right)^{2}}{2}}~{};$
$\displaystyle\Delta\,p_{2}=\sqrt{\braket{p_{2}^{2}}\,-\,\braket{p_{2}}^{2}}\,=\,\dfrac{\sqrt{m+n+1}}{a\,\rho}\sqrt{\dfrac{a^{2}+\rho^{2}\left(\dot{\rho}-2\rho\,d\right)^{2}}{2}}~{}.$
(86)

The equal uncertainty products among the parameters $x_{i}$ and $p_{i}$, where
$i\,=\,[1,2]$, are therefore given by,

$\displaystyle\Delta\,x_{1}\,\Delta\,y_{1}=\,\left(m+n+1\right)\dfrac{\rho^{2}}{2}~{}$
$\displaystyle,~{}\Delta\,p_{1}\,\Delta\,p_{2}=\dfrac{(n+m+1)}{2}\left[\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right]\,~{},$
$\displaystyle\Delta\,x_{1}\,\Delta\,p_{1}=\Delta\,x_{2}\,\Delta\,p_{2}$
$\displaystyle=\dfrac{m+n+1}{2\,a}\sqrt{a^{2}+\rho^{2}\left(\dot{\rho}-2\rho\,d\right)^{2}}~{}.$
(87)

The uncertainties in the product mentioned above yield the values reported in
[1] as the parameter $d(t)$ tends to zero.

#### Equal uncertainty products for noncommutative operators :

Organizing the quantities derived in Eqn(s).(47, 49) based on the coordinate
mapping relation presented in Eqn.(5), we can determine that

$\displaystyle\braket{X_{i}}=0~{};~{}\braket{P_{i}}=0~{}~{}~{};~{}~{}~{}i\in[1,2]~{}.$
(88)

Now doing the same thing for the quantities derived in Eqn(s).(47, 67, 75, 78,
80) we establish that

$\displaystyle\braket{X_{i}^{2}}=\dfrac{(m+n+1)}{2}\,\left[\rho^{2}\,\left(1-\dfrac{\theta\,\Omega}{4}\right)+\dfrac{\theta^{2}}{4}\,\left(\dfrac{1}{\rho^{2}}+\dfrac{\left(\dot{\rho}-2\rho\,d\right)^{2}}{a^{2}}\right)-\dfrac{\theta\sqrt{-\theta\,\Omega\,}\rho}{2\,a}\left(\dot{\rho}-2\rho\,d\right)\right]-(m-n)\,\dfrac{\theta}{2}~{},$
(89)
$\displaystyle\braket{P_{i}^{2}}=\dfrac{(m+n+1)}{2}\,\left[\,\left(1-\dfrac{\theta\,\Omega}{4}\right)\left(\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right)+\dfrac{\Omega^{2}\rho^{2}}{4}\,+\dfrac{\Omega\sqrt{-\theta\,\Omega\,}\rho}{2\,a}\left(\dot{\rho}-2\rho\,d\right)\right]-(m-n)\,\dfrac{\Omega}{2}~{}.$
(90)

With these values in hand, we now proceed to determine the product of
variances for the NC operators. This leads to

$\displaystyle\Delta\,X\,\Delta\,Y=\sqrt{\braket{X^{2}}\braket{Y}^{2}}=\braket{X^{2}}=\braket{Y^{2}}$
$\displaystyle=\dfrac{(m+n+1)}{2}\,\left[\rho^{2}\,\left(1-\dfrac{\theta\,\Omega}{4}\right)+\dfrac{\theta^{2}}{4}\,\left(\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right)-\dfrac{\theta\sqrt{-\theta\,\Omega\,}\rho}{2\,a}\left(\dot{\rho}-2\rho\,d\right)\right]-(m-n)\,\dfrac{\theta}{2}~{},$
(91)
$\displaystyle\Delta\,P_{X}\,\Delta\,P_{Y}=\sqrt{\braket{P_{X}^{2}}\braket{P_{Y}}^{2}}=\braket{P_{X}^{2}}=\braket{P_{Y}^{2}}$
$\displaystyle=\dfrac{(m+n+1)}{2}\left[\,\left(1-\dfrac{\theta\,\Omega}{4}\right)\left(\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right)+\dfrac{\Omega^{2}\rho^{2}}{4}+\dfrac{\Omega\sqrt{-\theta\Omega\,}\rho}{2\,a}\left(\dot{\rho}-2\rho\,d\right)\right]-(m-n)\dfrac{\Omega}{2}~{},$
(92)
$\displaystyle\Delta\,X\,\Delta\,P_{X}=\Delta\,Y\,\Delta\,P_{Y}=\sqrt{\braket{X^{2}}\braket{P_{X}^{2}}}=\sqrt{\braket{Y^{2}}\braket{P_{Y}^{2}}}$
$\displaystyle=\left\\{\dfrac{(m+n+1)}{2}\,\left[\rho^{2}\,\left(1-\dfrac{\theta\,\Omega}{4}\right)+\dfrac{\theta^{2}}{4}\,\left(\dfrac{1}{\rho^{2}}+\dfrac{\left(\dot{\rho}-2\rho\,d\right)^{2}}{a^{2}}\right)-\dfrac{\theta\sqrt{-\theta\,\Omega\,}\rho}{2\,a}\left(\dot{\rho}-2\rho\,d\right)\right]-(m-n)\,\dfrac{\theta}{2}\right\\}^{\dfrac{1}{2}}$
$\displaystyle\times\left\\{\dfrac{(m+n+1)}{2}\,\left[\,\left(1-\dfrac{\theta\,\Omega}{4}\right)\left(\dfrac{1}{\rho^{2}}+\dfrac{(\dot{\rho}-2\rho\,d)^{2}}{a^{2}}\right)+\dfrac{\Omega^{2}\rho^{2}}{4}\,+\dfrac{\Omega\sqrt{-\theta\,\Omega\,}\rho}{2\,a}\left(\dot{\rho}-2\rho\,d\right)\right]-(m-n)\,\dfrac{\Omega}{2}\right\\}^{\dfrac{1}{2}}~{}.$
(93)

It is nice to note that our results are consistent with those found in [1] as
the parameter $d$ and the multiplication of the NC parameters $\theta\Omega$
tend to zero.

## 6 Conclusion

In this work, our main focus is to investigate the behaviour of a NC harmonic
oscillator in time dependent background, as studied in [1], when it is mapped
in terms of commutative variables using a generalized version of the standard
Bopp-shift relations, recently introduced by [2]. Specifically, we aim to
investigate how this approach modifies the behaviours of the oscillator
undergoing mapping via the standard Bopp-shift relations in [1] . For doing
so, a two-dimensional harmonic oscillator is considered in NC framework with
time-dependent NC parameters. To facilitate our analysis, we employ the
generalized version of the standard Bopp-shift relation [2] to transform the
system into commutative variables. Then we present the exact eigenfunction
expression of the Hamiltonian, previously derived in [45], which addresses a
similar time-dependent system using the Lewis invariant associated with the
non-linear Ermakov-Pinney (EP) equation . Considering the potential benefits
of solving the EP equation, which acts as a constraint relation among the
Hamiltonian parameters, we proceed to explicitly solve this equation. As our
work is an extended version of the system described in [1], where the exact
analytical solutions of the EP equation were found by following the Chiellini
integrability condition [48], we expand upon their solution set. We obtained
the explicit value of the additional EP parameter that corresponds to the set
of values of the EP parameters derived in [1]. Here we observed some
interesting features about the newly derived EP parameter both in exponential
and rational form. In the exponential solution set, while the first three EP
parameters, found in [1], have distinct individual form, the fourth one
appears with a generic structure. This structure allows for a special form of
the solution to be derived by setting a suitable value for a certain constant
present in the general form of the parameter. We also provided an example
illustrating the emergence of this special functional form for the fourth EP
parameter. In the case of rationally varying EP solution set, the power of
variation for the first three EP parameters, as provided in [1], is not fixed,
as it depends on a positive valued, real constant. However, the fourth EP
parameter just varies inversely with respect to time. We then verify that both
the EP solution set in expanded form, which varies exponentially and
rationally with respect to time, is consistent with the Chiellini
integrability condition. Next, we compute the expectation value of the
Hamiltonian in a generic form and show that, in the absence of the modified
term from the coordinate mapping relations, it reproduces the result obtained
in our previous communication [43], where we used the standard Bopp-shift
relations to map a system of damped harmonic oscillator from NC space to
commutative space. The generic form of the energy expectation value in the
eigenstates of the Hamiltonian, for a particular choice of quantum number, is
explicitly explored through the explicit exponential and rational form of the
EP solution set. In addition to creating a graphical representation of the
explicit energy expression for both the rationally varying and exponentially
varying cases, we also compared each case graphically with its behaviour
associated with the standard Bopp-shift relations. Initially, the dynamics of
the exponential energy expression shows an increase with increase with time
and then saturates. Throughout this process, the associated behaviour governed
by the standard Bopp-shift relations remains constant over time. In contrast,
the dynamics of the rational energy expression, containing different
coefficients of variations, gradually decrease to zero over time. In the final
part of our work, we computed variances for both commutative and NC operators,
and used these values to determine the generalized form of uncertainty
equality relations between them. Expectedly, the equality relations reproduce
the results obtained in [1] in appropriate limits. Although the inequality
relations would complete the analysis, we would like to report them in future
work.

## Acknowledgement

MD would like to thank Mr. Soham Sen for having useful discussion with him to
carry out this work and for his helpful assistance in operating the software
Mathematica.

## Appendix A: Associated Laguerre polynomial identity

In this Appendix, we shall prove the following identity involving the
associated Laguerre polynomial.

$\displaystyle(n-m+1)\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+1}_{m-1}(z)\,dz\,-\,\int_{0}^{\infty}\,z^{n-m+1}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+2}_{m-2}(z)\,dz=0~{}.$
(94)

We begin by describing another identity related to the associated Laguerre
polynomial. The identity reads

$\displaystyle\,z\,L^{n-m+1}_{m-1}(z)\,=\,n\,L^{n-m}_{m-1}(z)-\,m\,L^{n-m}_{m}(z)~{}.$
(95)

We can express the left-hand side of Eqn.(94) in the following manner,

$\displaystyle(n-m+1)\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+1}_{m-1}(z)\,dz\,-\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,\left[z\,L^{n-m+2}_{m-2}(z)\right]\,dz=0~{}.$
(96)

As per the identity in Eqn.(95),

$\displaystyle
z\,L^{n-m+2}_{m-2}(z)\,=\,n\,L^{n-m+1}_{m-2}(z)-\,(m-1)\,L^{n-m+1}_{m-1}(z)~{}.$
(97)

By substituting the above identity into Eqn.(96), and performing some
algebraic calculations, we obtain

$\displaystyle(n-m+1)\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+1}_{m-1}(z)\,dz-\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,\left[z\,L^{n-m+2}_{m-2}(z)\right]\,dz$
$\displaystyle=n\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,\left[\,L^{n-m+1}_{m-1}(z)-L^{n-m+1}_{m-2}(z)\,\right]\,dz~{}.$
(98)

Applying the second relation in Eqn.(59), the expression above can be
simplified as,

$\displaystyle
n\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,\left[\,L^{n-m+1}_{m-1}(z)-L^{n-m+1}_{m-2}(z)\,\right]\,dz=n\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m}_{m-1}(z)\,dz~{}.$
(99)

Now the right hand side of the above relation turns out to be zero as per the
orthonormality condition mentioned in Eqn.(59). Therefore, we have

$\displaystyle(n-m+1)\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+1}_{m-1}(z)\,dz\,$
$\displaystyle-\,\int_{0}^{\infty}\,z^{n-m+1}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m+2}_{m-2}(z)\,dz$
$\displaystyle=n\,\int_{0}^{\infty}\,z^{n-m}\,e^{-z}\,L^{n-m}_{m}(z)\,L^{n-m}_{m-1}(z)\,dz$
$\displaystyle=0~{}.$ (100)

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