# The Evolution of Substance Use Coverage in the Philadelphia Inquirer

Layla Bouzoubaa, 1 Ramtin Ehsani, 1 Preetha Chatterjee, 1 Rezvaneh Rezapour 1

###### Abstract

The media’s representation of illicit substance use can lead to harmful
stereotypes and stigmatization for individuals struggling with addiction,
ultimately influencing public perception, policy, and public health outcomes.
To explore how the discourse and coverage of illicit drug use changed over
time, this study analyzes 157,476 articles published in the Philadelphia
Inquirer over a decade. Specifically, the study focuses on articles that
mentioned at least one commonly abused substance, resulting in a sample of
3,903 articles. Our analysis shows that cannabis and narcotics are the most
frequently discussed classes of drugs. Hallucinogenic drugs are portrayed more
positively than other categories, whereas narcotics are portrayed the most
negatively. Our research aims to highlight the need for accurate and inclusive
portrayals of substance use and addiction in the media.

## Introduction

The portrayal of illicit substance use in the media has long been a topic of
concern, given the potential impact it can have on public perception and
policy. Previous studies highlighted the harmful effects of stigmatizing
coverage on individuals struggling with substance use disorder (SUD) and the
potential for biased and inaccurate media coverage to perpetuate harmful
stereotypes and stigma (Johnston, O’malley, and Bachman 2004; Caburnay et al.
2003; Hughes, Lancaster, and Spicer 2011). Moreover, media coverage often
prioritizes criminal justice actions over critical health concerns associated
with substance use (Hayden Griffin et al. 2023). This bias has been further
observed in the case of designer drugs such as “bath salts,” which were
portrayed as an “epidemic” with harmful effects while neglecting relevant
clinical research and mental health issues associated with them (Swalve and
DeFoster 2016). As a result, health legislation was inadequately informed, and
underlying issues related to substance use are not addressed.

Given the potential impact of media coverage, it is essential to carefully
examine how the media frames discussions about illicit drug use and how that
framing changes over time. In this study, we examine news articles from the
Philadelphia Inquirer to understand changes in media coverage of illicit
substance use over time. Philadelphia was chosen because it has a high rate of
overdose deaths (40 per 100K) and is home to a significant open-air drug
market (Niamatullah, Auchincloss, and Livengood 2022). Analyzing local media
coverage of SUD can reveal insights on its portrayal, identify reporting gaps,
and inform policymaking and public education. This type of study can also
track the policy process and evolving relationships between actors, like
advocacy groups or healthcare professionals, over time (Paalman 1997). More
specifically, we analyze the portrayal of nine main drug classes, including
stimulants, narcotics, cannabis, hallucinogens, depressants, designer drugs,
drugs of concern, treatment, and miscellaneous over a period of ten years and
across 157,476 published articles. The study then focuses on approximately
3,903 articles that mention at least one commonly abused substance and applies
various natural language processing (NLP) techniques to extract drug-related
themes over time. We aim to answer the following research questions:

  * -

RQ1: How do the occurrences of various drug classes change over time, and in
what contexts are different substances co-mentioned?
We use a list of commonly abused drugs, map them to drug classes, extract
articles with at least one mention of drugs, and use TF-IDF to find salient
words in the subsets of articles discussing co-occurring drug classes.

  * -

RQ2: How has media coverage of illicit substance use evolved over time?
Dynamic topic modeling and sentiment analysis are employed to capture the
shift of prevalent themes and tone with respect to drug classes, over time.

Our analysis indicates a shift in the discussion of drug-related topics, with
cannabis being the most frequently discussed drug class, followed by
narcotics. Articles on narcotics and treatment drugs frequently co-occur,
focusing primarily on criminality, legislature, and overdose. News articles
discussing cannabis, hallucinogens, and treatment drugs tend to be positive
while those about narcotics, stimulants, and depressants are negative. The
insights gained from this work can inform better public health messaging,
affect drug policy decisions made by lawmakers, and enable medical
professionals to create more targeted patient education and awareness programs
while promoting the principles of harm reduction.

## Methodology

### Data Collection

We utilize ProQuest for data collection and accessing news articles. ProQuest
is an online platform consisting of thousands of databases that provide access
to a diverse set of publications, including journals and newspapers. In order
to collect data, it was necessary to log in to the ProQuest database using our
individualized credentials, after which we were able to extract/scrape the
data. We collect a total of 157,476 news articles from the Philadelphia
Inquirer, covering the period from January 1, 2013, to December 31, 2022. It’s
important to note that our dataset may not represent the complete set of
articles published by the Philadelphia Inquirer, as ProQuest’s database may
not be comprehensive. Our data comprises the complete text of news articles,
alongside several metadata attributes such as date, author, title, links, and
subject keywords.111The meta-data of news articles, list of drugs, and code
can be found on https://github.com/social-nlp-lab/drugs˙in˙inquirer

### Drug Name Extraction

We utilize the categorization provided by the National Institute on Drug Abuse
(NIDA), which comprises 28 drug categories, each with its corresponding
commercial names. To minimize irrelevant articles, we manually verified drug
names and removed ambiguous terms with multiple meanings, e.g., the term “pot”
could refer to “flowering pot” or “cooking pot” in addition to the drug
“marijuana”. We then assign each drug to one of the nine classes established
by the Drug Enforcement Administration (DEA) (DEA 2020). These classes
include:

  * •

Cannabis (C): marijuana is a mind-altering (psychoactive) drug, produced by
the cannabis sativa plant.“Cannabis” and “marijuana” are often used
interchangeably, resulting in the appearance of cannabis as both the drug
class and the drug name.

  * •

Depressants (D): known to induce sleep, relieve anxiety and muscle spasms, and
prevent seizures, e.g., barbiturates, and sedative-hypnotic substances like
GHB.

  * •

Designer Drugs (DD): produced illicitly with a slightly altered chemical
structure to mimic the pharmacological effects of controlled substances, e.g.,
synthetic marijuana or synthetic cathinones.

  * •

Drugs of Concern (DC): unregulated drugs that can be harmful if abused, e.g.,
kratom and xylazine.

  * •

Hallucinogens (H): derived from plants and fungi and renowned for their
capacity to modify human perception and mood, e.g., LSD, mushrooms, and
ecstasy.

  * •

Narcotics (N): refers to opium, opium derivatives, and their semi-synthetic
substitutes. “Opioid” is a more current and precise term to describe these
drugs, e.g., heroin, OxyContin, codeine, morphine, and fentanyl.

  * •

Stimulants (S): drugs accelerating body’s functions, e.g., methamphetamine,
cocaine, and amphetamines.

  * •

Treatment (T): substances aiding the treatment of opioid addiction, e.g.,
methadone, Suboxone, and naloxone.

  * •

Miscellaneous (M): substances that can be abused but don’t belong to any
classes, e.g., steroids.

| C | D | DD | DC | H | N | S | T | Total
---|---|---|---|---|---|---|---|---|---
2013 | 152 | 23 | 3 | 2 | 38 | 142 | 108 | 16 | 484
2014 | 150 | 10 | 0 | 1 | 53 | 117 | 64 | 21 | 416
2015 | 132 | 6 | 0 | 0 | 35 | 117 | 49 | 30 | 369
2016 | 152 | 12 | 0 | 0 | 44 | 132 | 63 | 43 | 446
2017 | 203 | 12 | 0 | 0 | 37 | 223 | 53 | 78 | 606
2018 | 226 | 14 | 8 | 6 | 43 | 228 | 73 | 98 | 696
2019 | 234 | 10 | 4 | 2 | 55 | 148 | 90 | 48 | 591
2020 | 119 | 7 | 0 | 0 | 18 | 69 | 48 | 31 | 292
2021 | 124 | 7 | 0 | 0 | 40 | 86 | 41 | 41 | 339
2022 | 145 | 5 | 1 | 0 | 31 | 99 | 38 | 43 | 362
Total | 1637 | 106 | 16 | 11 | 394 | 1361 | 627 | 449 |

Table 1: Number of drug class occurrences per year, with bold values
indicating the most frequent drug class per year. Drug classes include C
(Cannabis), D (Depressants), DD (Designer Drugs), DC (Drugs of Concern), H
(Hallucinogens), N (Narcotics), S (Stimulants), and T (Treatment).

Figure 1: Number of substances per drug class in our dataset.

Figure 2: News topic evolution over time, with each line representing a
cluster of related topics.

Using our mapped list of drug names, we identify news articles that mention
any of these drugs in either their title or body. Subsequently, we group these
articles into clusters based on the respective drug classes. If an article
mentions drugs from multiple classes, we assign it to all relevant drug
classes. The number of articles per drug class and year is shown in Table 1.
Out of a total of 157,476 articles analyzed, 3,903 referenced at least one
drug class within their text.

In addition to assessing the frequency of drugs, we also examine co-
occurrences of drugs in the same articles each year. To better understand
these co-occurrences and the surrounding contexts, we extract the most
significant words based on their TF-IDF scores from a subset of articles where
drugs most frequently co-occurred.

### Topic Modeling

To investigate the evolution of drug-related articles, we employ dynamic topic
modeling (Blei and Lafferty 2006) to understand the distribution of topics
over time. We use BERTopic as it can preserve the semantic structure of texts
(Grootendorst 2022). Since BERTopic operates on a bag-of-words representation
of the text (Grootendorst 2022), we preprocess the text by converting it to
lowercase, removing URLs, extending contractions, and eliminating stopwords.
This minimizes noise and dimensionality, allowing our models to recognize
important patterns and topics. It is important to note that BERT has a token
limit of 512, while the median length of news articles in our data is 749
tokens with a standard deviation of 573. News articles’ opening paragraphs
contain the most important information, therefore, we assume the first 512
tokens of news articles in our data represent the key points.

### Tone Detection

VADER (Hutto and Gilbert 2014), a validated rule-based model for sentiment
analysis, is used to analyze shifts in the tone of news articles that center
on substance use. We calculate the average sentiment scores of articles in
six-month periods using VADER’s compound score, allowing us to monitor changes
in sentiment over time.

## Results

### How do the occurrences of various drug classes change over time, and in
what contexts are different substances co-mentioned?

The results show a significant shift in the discussion of illicit substance
use over the past decade. As shown in Table 1, cannabis is the most frequently
discussed drug class, followed by narcotics. The annual count of news articles
indicates that 2018 saw the highest frequency of drug-related mentions in news
coverage. In 2018, several significant events occurred, including a former
player for the Philadelphia Eagles admitting to recreational cannabis use
during his career (Hermann 2019), Philadelphia Mayor Jim Kenney advocating for
the legalization of recreational cannabis use (Maas 2018), and the City of
Philadelphia hosting its first-ever Cannabis Opportunity Conference (Brown
2018). The frequency of cannabis in the news doesn’t necessarily imply that it
is used more than other substances, as it could be affected by the prevailing
social and political environment. Our findings suggest that the majority of
articles mentioning cannabis are centered around the legality of its use.
Narcotics (i.e., opioids) are the second most discussed drug class, and also
the most prevalent in 2017 and 2018, indicating the severity of discussions
and possibly the opioid crisis during those years. Philadelphia experienced
some of the highest rates of unintentional overdose rates, and the synthetic
opiate fentanyl became increasingly fatal (DOH 2022). As shown in Table 1,
designer drugs and drugs of concern are mentioned in only 16 and 11 articles,
respectively. Due to the sparse data on these topics in our corpus, we exclude
them from further analysis.

Further analysis of drug counts per class shows that cannabis is primarily
associated with marijuana (Figure 1). Despite the controversy surrounding the
use of the term “marijuana” (Halperin 2018), there has been a consistent and
steady usage of this word in news articles over the years. In contrast,
depressant drugs like Xanax and Ambien are more frequently mentioned together,
likely due to their widespread use for anxiety and sleep disorders and their
potential for abuse and addiction. Similarly, cocaine and acid are the most
frequently mentioned drugs in the stimulant and hallucinogen drug classes,
respectively, indicating their enduring popularity and widespread use. It is
worth noting that our temporal analysis of drug mentions uncovers patterns in
drug discussion over time, which can be influenced by a variety of factors,
such as changes in drug policies and cultural shifts in attitudes towards
various elements of drug use, like policy, treatment, and research.

| Drug 1 | Drug 2 | N |  | Drug 1 | Drug 2 | N
---|---|---|---|---|---|---|---
2013 | heroin | methadone | 52 | 2018 | fentanyl | cocaine | 270
heroin | cocaine | 45 | heroin | cocaine | 194
barbiturates | cannabis | 16 | heroin | marijuana | 127
2014 | heroin | naloxone | 42 | 2019 | heroin | marijuana | 115
marijuana | cocaine | 36 | heroin | naloxone | 77
heroin | cocaine | 34 | heroin | cocaine | 71
2015 | heroin | marijuana | 40 | 2020 | heroin | methadone | 198
heroin | methadone | 19 | heroin | buprenorphine | 81
heroin | meth | 11 | fentanyl | cocaine | 53
2016 | heroin | marijuana | 51 | 2021 | buprenorphine | cannabis | 108
heroin | cocaine | 45 | heroin | cannabis | 78
heroin | naloxone | 44 | heroin | marijuana | 41
2017 | heroin | methadone | 157 | 2022 | buprenorphine | cannabis | 351
heroin | cocaine | 90 | buprenorphine | marijuana | 153
heroin | naloxone | 80 | fentanyl | cocaine | 114

Table 2: The most frequent co-occurred drugs per year.

The co-occurrence analysis (Table 2) revealed that certain substances are
frequently discussed together, indicating a connection between their use,
among other possible associations. Heroin was once the most discussed
substance in Philadelphia until it was surpassed by fentanyl in 2018. This is
in line with reports of adulteration and an increase in unintentional overdose
deaths containing fentanyl in Philadelphia (Drug Enforcement Administration
2021). Stimulants, like cocaine, and cannabis are commonly observed to co-
occur with narcotics, and there is growing concern among the public and
policymakers over an increasing number of unintentional overdose deaths
involving stimulants. Marijuana and heroin are frequently mentioned together
in discussions around legalization and concerns about adulteration from
fentanyl. Our analysis also shows that narcotics and treatment drugs,
including methadone, naloxone, and buprenorphine, are the second most commonly
co-located drugs in articles. These articles primarily focus on criminality,
legislature, and overdose, as determined by our TF-IDF analysis. There is
growing concern over the co-occurrence of stimulants, like cocaine, with
narcotics, like heroin, due to the rising number of fatalities associated with
their use (Johnson et al. 2021). The Philadelphia Inquirer has been
particularly active in discussing the “opioid epidemic,” with a focus on
treatment. The use of treatment drugs like methadone and naloxone is seen as a
positive and proactive response to drug addiction (National Academies of
Sciences et al. 2019). Buprenorphine, another popular substance used to treat
opioid use disorder, gained attention in 2022 after President Biden eliminated
the restriction on the number of prescriptions providers could issue per
month, making it more accessible compared to methadone (Pew 2021).

Figure 3: Drug class sentiment scores over the past decade.

### How has media coverage of illicit substance use evolved over time?

Utilizing dynamic topic modeling with BERTopic, we were able to generate a
diverse set of topics covering various themes. The review of the initial
results revealed a number of overlapping topics. Therefore, we iteratively
refined the model by decreasing the number of topics until we achieved a high-
quality output. Figure 2 presents temporal changes of topic clusters in our
data. The result shows that cannabis-related topics are the most prevalent in
our dataset, with a specific focus on legislation. We also observe a
significant number of news articles discussing drug use in films and
television series, indicating the possible role of popular media in shaping
public perceptions and attitudes toward drugs and social issues (Rezapour and
Diesner 2017; Diesner, Rezapour, and Jiang 2016).

The sentiment analysis of news articles demonstrates that the tone of news
articles varies across drug classes and evolves over time (Figure 3). News
articles with hallucinogenic drugs tend to have a more positive tone compared
to other classes. The positive tone may be attributed to recent studies on the
use of psychedelic drugs for the treatment of mental health disorders such as
treatment-resistant depression and post-traumatic stress disorder. For
instance, we identified an article published in the Philadelphia Inquirer
discussing the use of LSD and shrooms to treat PTSD (Ao 2020). This article
reflects the positive sentiment toward the therapeutic potential of
hallucinogenic drugs. Furthermore, the majority of news articles mentioning
cannabis, hallucinogens, and treatment drugs exhibit positive sentiments,
whereas news articles about narcotics, stimulants, and depressants depict
negative sentiments. The sentiment scores assigned to each article range from
-1 (most negative) to +1 (most positive), and neutral sentiments are rarely
observed in our dataset

Overall, our analysis indicates a shift in the discussion surrounding drug-
related topics, especially hallucinogens, towards a more positive view of
medicated-assisted treatment and novel applications of these substances. This
emphasizes the importance of reporting drug-related incidents using harm
reduction principles to encourage safer and managed use. Further research is
necessary to fully understand the context and implications of these findings.
It is also important to acknowledge that our analysis only captures a subset
of the wider discussion surrounding these drugs and that additional studies
may reveal further insights.

## Conclusion and Future Work

Over the past decade, there has been an evolution in how the media portrays
illicit substance use, as our study has revealed. We analyzed news articles
published in the Philadelphia Inquirer between 2013 and 2022 and found that
cannabis was the most frequently discussed drug class, followed by narcotics.
News articles about hallucinogenic drugs tend to have a more positive tone
compared to other categories of drugs, while articles on narcotics were the
most negative. By examining changes in the tone and frequency of drug-related
discussions over time, we can gain a better understanding of how societal
attitudes towards drugs and drug policies are evolving, and how this may
impact public health and well-being. It is important to note that our study
focused solely on news articles published in the Philadelphia Inquirer, and
therefore our findings are specific to this source and its coverage of
substance use within the Philadelphia and United States. Further research is
needed to explore how the portrayal of substance use in the media varies
across different regions and cultures.

## References

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# Exact solution of damped harmonic oscillator with a magnetic field in a time
dependent noncommutative space

Manjari Duttaa, Shreemoyee Ganguly b, Sunandan Gangopadhyay c
a,c Department of Theoretical Sciences,
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
manjaridutta<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we have obtained the exact eigenstates of a two dimensional
damped harmonic oscillator in the presence of an external magnetic field
varying with respect to time in time dependent noncommutative space. It has
been observed that for some specific choices of the damping factor, the time
dependent frequency of the oscillator and the time dependent external magnetic
field, there exists interesting solutions of the time dependent noncommutative
parameters following from the solutions of the Ermakov-Pinney equation.
Further, these solutions enable us to get exact analytic forms for the phase
which relates the eigenstates of the Hamiltonian with the eigenstates of the
Lewis invariant. Then we compute the expectation value of the Hamiltonian. The
expectation values of the energy are found to vary with time for different
solutions of the Ermakov-Pinney equation corresponding to different choices of
the damping factor, the time dependent frequency of the oscillator and the
time dependent applied magnetic field. We also compare our results with those
in the absence of the magnetic field obtained earlier.

##  1 Introduction

The Landau problem of a charged particle moving in a two dimensional plane
under the influence of a magnetic field acting perpendicular to the plane has
been looked upon by physicists over the years, not only from the pedagogical
interest of formation of discrete energy levels known as Landau levels, but
also due to the multifaceted applications of this problem. In this context, an
extremely intriguing problem is that of a charged oscillator placed in a
magnetic field, acting perpendicular to the plane in which it is oscillating.
In addition, one may also consider an electric field lying along the plane of
oscillation. The eigenfunction and the eigenvalues of a charged particle in a
magnetic field in the presence of a time-dependent background electric field
with time-dependent mass and frequency have been determined in [1]. The
problem becomes even more fascinating when one places the Landau oscillator in
a noncommutative phase space. This problem has been looked upon by an earlier
study [2] in the presence of a time dependent magnetic field. The system there
was considered in a noncommutative space. The simplest setting of
noncommutative [NC] space is a two dimensional quantum mechanical space in
which one replaces the standard set of commutation relations between the
canonical coordinates by NC commutation relations $[X,Y]=i\theta$, where
$\theta$ is a positive real constant. Study of quantum mechanical systems in
such NC space has allured theoretical physicists since the work by Synder [3].
Since then the necessity for NC spaces has been established to ensure the
attainment of gravitational stability [4] in the present theories of quantum
gravity, namely, string theory [5, 6] and loop quantum gravity [7]. This has
triggered several studies on quantum mechanical systems in such spaces in the
literature [8]-[18].

However, the mentioned study [2] on charged oscillators in the presence of
magnetic field, considers only noncommutativity amongst position variables.
So, we in our present communication extend the study to a space where
noncommutativity exists not only amongst spatial variables but also amongst
momentum variables. Also unlike the previous study we have considered the NC
space to be time dependent. Moreover, our oscillator is considered to be
damped by an explicit damping factor in order to model a realistic situation.
A damped oscillator in two dimensional NC space has been studied by us in an
earlier communication [19]. Before our communication one of the very few works
which had studied damped quantum harmonic oscillators in two dimensional space
was the work by Lawson et.al. [20]. We extended their model to a two
dimensional NC space. But at present our objective is to study how the
interplay of damping and an external time dependent magnetic field modulates
the energetics of a charged oscillator in a time dependent NC space where
spatial as well as momentum noncommutativity is present.

Our present study is one of the very first to study a damped quantum harmonic
oscillator with time varying frequency in two dimensional NC space in the
presence of a time-dependent magnetic field. In our earlier study [19] we had
seen that the expectation value of energy of an oscillator decays due to
damping even in NC space. In the present study we intend to investigate the
change in energetics of the damped quantum oscillator in NC space under the
influence of magnetic field having various kinds of time dependence. For this
purpose we first set up the Hamiltonian of the damped quantum harmonic
oscillator in two dimensional NC space under the presence of a time varying
magnetic field and then express it in terms of commutative variables. This is
done in Section 2. After that we solve the Hamiltonian using the method of
invariants [21, 22, 23] in Section 3. It must be noted that the eigenfunction
of the said Hamiltonian is a product of the eigenfunction of the invariant and
a phase factor. Both the eigenfunction and phase factor are expressed in terms
of time dependent parameters which obey the non-linear differential equation
known as Ermakov-Pinney (EP) equation [24, 25]. Next in Section 4 we choose
the parameters of the damped system in the presence of magnetic field in such
a way that they satisfy all the equations representing the system as well as
provide us exact closed form solutions for the system for various choices of
time variation of the applied field. In Section 5 we calculate the expectation
value of energy of the damped oscillator explicitly and graphically explore
how the time dependence of the magnetic field alters the time evolution of the
energetics of the damped quantum system having a time varying frequency of
oscillation in two dimensional NC space.

##  2 Model of the two-dimensional harmonic oscillator in magnetic field

In our study the system that we consider is a combination of two non-
interacting damped harmonic oscillators affected by a time dependent magnetic
field in two dimensional NC space. Both the oscillators have equal time
dependent frequencies, coefficients of friction, equal mass and equal charge
in NC space. Such a model of two-dimensional Landau problem of harmonic
oscillator in a time dependent magnetic field was considered in an earlier
communication [2] in spatially noncommutative configuration space. In this
work, however, we extend the model by considering the system in time dependent
NC space. Also it must be noted that the noncommutativity we consider is not
restricted to spatial variables like the earlier studies but also extends to
momentum noncommutativity.111We shall be considering NC phase space in our
work. However, we shall be generically referring to this as NC space.

The Hamiltonian of the two dimensional oscillator in magnetic field has the
following form,

$H(t)=\dfrac{f(t)}{2M}\left[(P_{1}-qA_{1})^{2}+(P_{2}-qA_{2})^{2}\right]+\dfrac{M\omega^{2}(t)}{2f(t)}({X_{1}}^{2}+{X_{2}}^{2})~{}~{};$
(1)

where the damping factor $f(t)$ is given by,

$f(t)=e^{-\int_{0}^{t}\eta(s)ds}$ (2)

with $\eta(s)$ being the coefficient of friction and $A_{i}$, the vector
potential of a time dependent magnetic field $B(t)$ is chosen in Coulomb gauge
as,

$A_{i}=-\dfrac{B(t)}{2}\epsilon_{ij}X^{j}~{}~{};$ (3)

where $i,j=1,2$ and $\epsilon_{ij}=-\epsilon_{ji}$ with $\epsilon_{12}=1$.
Here $\omega(t)$ is the time dependent angular frequency of the oscillators ,
$M$ and $q$ are their mass and charge respectively. The position and momentum
coordinates $(X_{i},P_{i})$ are noncommuting variables in NC space, that is,
their commutators are $[X_{1},X_{2}]~{}\neq~{}0$ and
$[P_{1},P_{2}]~{}\neq~{}0$. The corresponding canonical variables
$(x_{i},p_{i})$ in commutative space are such that the commutator
$[x_{i},p_{j}]=i\hbar\delta_{i,j}$, $[x_{i},x_{j}]=0=[p_{i},p_{j}]$;
($i,j=1,2$).

In order to express the NC Hamiltonian in terms of the standard commutative
variables explicitly, we apply the standard Bopp-shift relations [26]
($\hbar=1$):

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

Here $\theta(t)$ and $\Omega(t)$ are the NC parameters for space and momentum
respectively, such that $[X_{1},X_{2}]~{}=i\theta(t)$,
$[P_{1},P_{2}]~{}=i\Omega(t)$ and
$[X_{1},P_{1}]=i[1+\frac{\theta(t)\Omega(t)}{4}]=[X_{2},P_{2}]$; ($X_{1}\equiv
X$, $X_{2}\equiv Y$, $P_{1}\equiv P_{x}$, $P_{2}\equiv P_{y}$).

The Hamiltonian in terms of $(x_{i},p_{i})$ coordinates is therefore given by
the following relation,

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

The time dependent coefficients in the above Hamiltonian are given as,

$\displaystyle
a(t)=\dfrac{f(t)}{M}+\dfrac{qB(t)f(t)\theta(t)}{2M}+\dfrac{1}{4}\left[\dfrac{q^{2}B^{2}(t)f(t)}{4M}+\dfrac{M\omega^{2}(t)}{f(t)}\right]\theta^{2}(t)$
(7) $\displaystyle
b(t)=\dfrac{q^{2}B^{2}(t)f(t)}{4M}+\dfrac{M\omega^{2}(t)}{f(t)}+\dfrac{qB(t)f(t)\Omega(t)}{2M}+\dfrac{f(t)\Omega^{2}(t)}{4M}$
(8) $\displaystyle
c(t)=\dfrac{1}{2}\left[\dfrac{qB(t)f(t)}{M}\left(1+\dfrac{\theta(t)\Omega(t)}{4}\right)+\dfrac{\Omega(t)f(t)}{M}+\left(\dfrac{q^{2}B^{2}(t)f(t)}{4M}+\dfrac{M\omega^{2}(t)}{f(t)}\right)\theta(t)\right]~{}.$
(9)

Here it must be noted that although our Hamiltonian given by Eqn.(6) has the
same form as that in [27] and [19] to study a system of a two dimensional
harmonic oscillator and damped harmonic oscillator in NC space, the time
dependent Hamiltonian coefficients (given by Eqn(s).(7),(8),(9)) have a
modified form. This is because our present system of damped harmonic
oscillator is studied in the presence of a time dependent magnetic field in
two-dimensional NC space. Thus, both the damping factor $f(t)$ and the
magnetic field $B(t)$ modulate and alter the Hamiltonian coefficients from the
form considered in earlier study [27] and [19] respectively. It is relevant to
mention that those coefficients also differ from those obtained in [2] as the
considered noncommutativity in that study is time independent and exists only
in the configuration space.

##  3 Solution of the model Hamiltonian

In order to find the solutions of the model Hamiltonian $H(t)$ (Eqn.(6))
representing the two-dimensional damped harmonic oscillator with magnetic
field in NC space, we follow the route suggested by Lewis et.al. [21] in their
work. First we construct the time-dependent Hermitian invariant operator
$I(t)$ corresponding to our Hamiltonian operator $H(t)$ (given by Eqn.(6)).
This is because if one can solve for the eigenfunctions of $I(t)$,
$\phi(x_{1},x_{2})$, such that,

$I(t)\phi(x_{1},x_{2})=\epsilon\phi(x_{1},x_{2})$ (10)

where $\epsilon$ is an eigenvalue of $I(t)$ corresponding to eigenstate
$\phi(x_{1},x_{2})$, one can obtain the eigenstates of $H(t)$,
$\psi(x_{1},x_{2},t)$, using the relation given by Lewis et. al. [21] which is
as follows,

$\psi(x_{1},x_{2},t)=e^{i\Theta(t)}\phi(x_{1},x_{2})$ (11)

where the real function $\Theta(t)$ which acts as the phase factor will be
discussed in details later.

###  3.1 The Time Dependent Invariant

Next, following the approach taken by Lewis et.al. [21], we need to construct
the operator $I(t)$ which is an invariant with respect to time, corresponding
to the Hamiltonian $H(t)$, as mentioned earlier, such that $I(t)$ satisfies
the condition,

$\dfrac{dI}{dt}=\partial_{t}{I}+\dfrac{1}{i}[I,H]=0.$ (12)

The procedure is to choose the Hermitian invariant $I(t)$ to be of the same
homogeneous quadratic form defined by Lewis et. al. [21] for time-dependent
harmonic oscillators. However, since we are dealing with a two-dimensional
system in the present study, $I(t)$ takes on the following form,

$I(t)=\alpha(t)({p_{1}}^{2}+{p_{2}}^{2})+\beta(t)({x_{1}}^{2}+{x_{2}}^{2})+\gamma(t)(x_{1}{p_{1}}+p_{2}{x_{2}}).$
(13)

Here we will consider $\hbar=1$ since we choose to work in natural units. Now,
using the form of $I(t)$ defined by Eqn.(13) in Eqn.(12) and equating the
coefficients of the canonical variables, we get the following relations,

$\displaystyle\dot{\alpha}(t)$ $\displaystyle=$ $\displaystyle-a(t)\gamma(t)$
(14) $\displaystyle\dot{\beta}(t)$ $\displaystyle=$ $\displaystyle
b(t)\gamma(t)$ (15) $\displaystyle\dot{\gamma}(t)$ $\displaystyle=$
$\displaystyle 2\left[\,b(t)\alpha(t)-\beta(t)a(t)\,\right]$ (16)

where dot denotes derivative with respect to time $t$.

To express the above three time dependent parameters $\alpha$,$\beta$ and
$\gamma$ in terms of a single time dependent parameter, we parametrize
$\alpha(t)=\rho^{2}(t)$. Substituting this in Eqn(s).(14, 16), we get the
other two parameters in terms of $\rho(t)$ as,

$\displaystyle\gamma(t)=-\dfrac{2\rho\dot{\rho}}{a(t)}~{}~{},~{}~{}\beta(t)=\dfrac{1}{a(t)}\left[\dfrac{{\dot{\rho}^{2}}}{a(t)}+{{\rho}^{2}}b+\dfrac{\rho\ddot{\rho}}{a(t)}-\dfrac{\rho\dot{\rho}\dot{a}}{a^{2}}\right].$
(17)

Now, substituting the value of $\beta$ in Eqn.(15), we get a non-linear
equation in $\rho(t)$ which has the form of the non-linear Ermakov-Pinney (EP)
equation with a dissipative term [27, 24, 25]. The form of the non-linear
equation is as follows,

$\ddot{\rho}-\dfrac{\dot{a}}{a}\dot{\rho}+ab\rho={\xi^{2}}\dfrac{a^{2}}{\rho^{3}}$
(18)

where ${\xi^{2}}$ is a constant of integration. This equation has similar form
to the EP equation obtained in [27], which is expected since our $H(t)$ has
the same form as theirs. However, once again it should be mentioned that the
explicit form of the time-dependent coefficients are different due to the
presence of the external magnetic field as well as the fact that the
oscillator is damped.

Now, using the EP equation we get a simpler form of $\beta$ as,

$\displaystyle\beta(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{a(t)}\left[\dfrac{{\dot{\rho}^{2}}}{a(t)}+\dfrac{{\xi^{2}}{a(t)}}{\rho^{2}}\right].$
(19)

Next, substituting the expressions of $\alpha$, $\beta$ and $\gamma$ in
Eqn.(13), we get the following expression for $I(t)$,

$I(t)=\rho^{2}({p_{1}}^{2}+{p_{2}}^{2})+\left(\dfrac{\dot{\rho}^{2}}{a^{2}}+\dfrac{{\xi^{2}}}{\rho^{2}}\right)({x_{1}}^{2}+{x_{2}}^{2})-\dfrac{2\rho\dot{\rho}}{a}(x_{1}{p_{1}}+p_{2}{x_{2}}).$
(20)

This form of the Lewis invariant in Cartesian coordinate is converted to polar
coordinate using the same procedure as followed in our previous communication
[19]. The invariant in polar coordinate takes the following form,

$\displaystyle
I(t)=\dfrac{\xi^{2}}{\rho^{2}}r^{2}+\left(\rho{p_{r}}-\dfrac{\dot{\rho}}{a}r\right)^{2}+\left({\dfrac{\rho{p_{\theta}}}{r}}\right)^{2}-\left({\dfrac{\rho\hbar}{2r}}\right)^{2}$
(21)

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

$\displaystyle
p_{r}=-i\left({\partial}_{r}+\dfrac{1}{2r}\right)~{}~{},~{}~{}p_{\theta}=-i{\partial_{\theta}}.$
(22)

Now we note from Eqn.(21) that the invariant $I(t)$ has the same form as that
used in [27] to study the undamped harmonic oscillator in NC space. The time-
dependent coefficients involved in the present study however differ due to the
presence of external magnetic field and damping in our system. Thus, we can
just borrow the expression of eigenfunction and the phase factors from [27]
for our present system.

###  3.2 Eigenfunction and phase factor

We depict the set of eigenstates of the invariant operator $I(t)$ as
$\ket{n,l}$, following the convention in [27]. Here, $n$ and $l$ are integers
such that $n+l\geqslant 0$. So we have the condition $l\geqslant-n$. Thus, if
$l=-n+m$, then $m$ is a positive integer; and the corresponding eigenfunction
in polar coordinate system has the following form (restoring $\hbar$),

$\displaystyle\phi_{n,m-n}(r,\theta)$ $\displaystyle=$
$\displaystyle\braket{r,\theta}{n,m-n}$ (23) $\displaystyle=$
$\displaystyle\lambda_{n}\dfrac{{(i\sqrt{\hbar}\rho)}^{m}}{\sqrt{m!}}r^{n-m}e^{i(m-n)\theta-\dfrac{a(t)-i\rho\dot{\rho}}{2a(t)\hbar{\rho}^{2}}r^{2}}U\left(-m,1-m+n,\dfrac{r^{2}}{\hbar\rho^{2}}\right)$
(24)

where $\lambda_{n}$ is given by

$\displaystyle\lambda_{n}^{2}=\dfrac{1}{\pi{n!}{(\hbar\rho^{2})}^{1+n}}~{}.$
(25)

Here, $U\left(-m,1-m+n,\dfrac{r^{2}}{\hbar\rho^{2}}\right)$ is Tricomi’s
confluent hypergeometric function [28, 29] and the eigenfunction
$\phi_{n,m-n}(r,\theta)$ satisfies the following orthonormality relation,

$\int_{0}^{2\pi}d\theta\int_{0}^{\infty}rdr\phi^{*}_{n,m-n}(r,\theta)\phi_{n^{{}^{\prime}},m^{{}^{\prime}}-n^{{}^{\prime}}}(r,\theta)=\delta_{nn^{{}^{\prime}}}\delta_{mm^{{}^{\prime}}}.$
(26)

Again following [27], the expression of the phase factor $\Theta(t)$ is given
by,

$\Theta_{\,n\,,\,l}(t)\,=\,(\,n\,+\,l\,)\,\int_{0}^{t}\left(c(T)-\dfrac{a(T)}{\rho^{2}(T)}\right)dT~{}.$
(27)

For a given value of $l=-n+m$, it would be given by [27],

$\Theta_{\,n\,,\,m\,-\,n\,}(t)=m\int_{0}^{t}\left(c(T)-\dfrac{a(T)}{\rho^{2}(T)}\right)dT~{}.$
(28)

We shall use this expression to compute the phase explicitly as a function of
time for various physical cases in the subsequent discussion.

The eigenfunction of the Hamiltonian therefore reads (using Eqn(s).(11, 24,
28))

$\displaystyle\psi_{n,m-n}(r,\theta,t)$ $\displaystyle=$ $\displaystyle
e^{i\Theta_{n,m-n}(t)}\phi_{n,m-n}(r,\theta)$ (29) $\displaystyle=$
$\displaystyle\lambda_{n}\dfrac{{(i\sqrt{\hbar}\rho)}^{m}}{\sqrt{m!}}\exp{\left[im\int_{0}^{t}\left(c(T)-\dfrac{a(T)}{\rho^{2}(T)}\right)dT\right]}$
$\displaystyle\times~{}r^{n-m}e^{i(m-n)\theta-\dfrac{a(t)-i\rho\dot{\rho}}{2a(t)\hbar{\rho}^{2}}r^{2}}U\left(-m,1-m+n,\dfrac{r^{2}}{\hbar\rho^{2}}\right).$

##  4 Solutions for the noncommutative damped oscillator
in magnetic field

In this communication we are primarily concerned about the evolution of the
solution due to the inclusion of a time dependent magnetic field in the
system. For this purpose we want to find the eigenfunctions of the
corresponding Hamiltonian due to interplay of damping and magnetic field. The
various kinds of damping in the presence of the applied magnetic field are
represented by various forms of the time dependent coefficients of the
Hamiltonian, namely, $a(t)$, $b(t)$ and $c(t)$. However, the various forms
must be constructed in such a way that they satisfy the non-linear EP equation
given by Eqn.(18). The procedure of this construction of exact analytical
solutions is based on the Chiellini integrability condition [30] and this
formalism was followed in [27]. So, in this communication for various forms of
$a(t)$ and $b(t)$, we get the corresponding form of $\rho(t)$ using the EP
equation together with the Chiellini integrability condition. In the
subsequent discussion we shall proceed to obtain solutions of the EP equation
for the damped oscillator in a magnetic field considered in NC space.

###  4.1 Solution Set-I for Ermakov-Pinney equation : Exponentially
decaying solutions

####  4.1.1 The Solution Set

The simplest kind of solution set of the EP equation under damping is the
exponentially decaying set used in [27]. The solution set is given by the
following relations,

$\displaystyle a(t)=\sigma e^{-\vartheta{t}}\,\,\,,\,\,\,b(t)=\Delta
e^{\vartheta{t}}\,\,\,,\,\,\rho(t)={\mu}e^{-\vartheta{t/2}}\,\,\,\,\,$ (30)

where $\sigma,\Delta$ and $\mu$ are constants. Here, $\vartheta$ is any
positive real number. Substituting the expressions for
$a(t),b(t)\,$and$\,\rho(t)$ in the EP equation, we can easily verify the
relation between these constants to be as follows,

$\mu^{4}=\dfrac{4\xi^{2}{\sigma^{2}}}{4\sigma\Delta-\vartheta^{2}}~{}.$ (31)

####  4.1.2 Study of the corresponding eigenfunctions

We now write down the eigenfunctions of the Hamiltonian for the chosen set of
time-dependent coefficients. For this purpose we need to choose explicit forms
of the damping factor $f(t)$ , angular frequency of the oscillator $\omega(t)$
and the applied magnetic field $B(t)$. The eigenfunction of the invariant
$I(t)$ (which is given by Eqn.(24)) takes on the following form for the
solution Set-I:

$\displaystyle\phi_{n,m-n}(r,\theta)=\lambda_{n}\dfrac{{(i{\mu}e^{-\vartheta{t/2}})}^{m}}{\sqrt{m!}}r^{n-m}e^{i(m-n)\theta-\dfrac{2\sigma+i\mu^{2}\vartheta}{4\sigma\mu^{2}{e^{-\vartheta{t}}}}r^{2}}U\left(-m,1-m+n,\dfrac{r^{2}{e^{\vartheta{t}}}}{\mu^{2}}\right)$
(32)

where $\lambda_{n}$ is given by

$\displaystyle\lambda_{n}^{\,2}\,=\,\dfrac{1}{\pi\,n!\,[\mu^{2}\,exp\,(-\vartheta{t})]^{1+n}}~{}.$
(33)

Next, we proceed to obtain explicit expressions of the phase factors for
various forms of the damping factor $f(t)$ and the angular frequency
$\omega(t)$ of the oscillator. The value of the applied magnetic field $B(t)$
is also tuned accordingly.
At first, we consider the most general form of damping factors and the applied
magnetic field which are given as follows,

$\displaystyle f(t)=e^{-\Gamma\,t}~{};~{}\omega(t)=\omega_{0}e^{-\delta
t/2}~{};~{}B(t)=B_{0}\,e^{\Lambda\,t}~{};$ (34)

where $\Gamma$ and $\delta$ are non-negative real constants and $\Lambda$ is
an arbitrary real constant. Substituting these relations in Eqn(s).(7, 8), we
get the most general form of the time dependent NC parameters as,

$\displaystyle\theta(t)=\dfrac{8Me^{-\Gamma\,t}}{q^{2}B_{0}^{2}e^{2(\Lambda-\Gamma)\,t}+4M^{2}\omega^{2}_{0}\,e^{-\delta\,t}}$
$\displaystyle\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma
e^{(2\Lambda-\Gamma-\vartheta)\,t}}{4M}+\omega_{0}^{2}\,e^{-\delta\,t}\left(M\sigma\,e^{(\Gamma-\vartheta)t}-1\right)}-\dfrac{qB_{0}e^{(\Lambda-\Gamma)\,t}}{2M}\right]$
(35) $\displaystyle\Omega(t)$
$\displaystyle=-qB_{0}\,e^{\Lambda\,t}+2e^{\Gamma\,t}\sqrt{M\Delta\,e^{(\vartheta-\Gamma)t}-M^{2}\omega_{0}^{2}\,e^{-\delta\,t}}~{}.$
(36)

In order to get the exact analytical form of the phase factor we choose some
suitable special forms of the constants $\vartheta$, $\Gamma$, $\delta$ and
$\Lambda$.

$\langle a\rangle$ Set-I , Case I

Here we set the constants

$\displaystyle\vartheta=\Gamma~{},~{}\delta=0~{},~{}\Lambda=0~{}.$ (37)

So, the parameters can be depicted by the following relations,

$\displaystyle
f(t)=e^{\,-\Gamma\,t}~{};~{}\omega(t)={\omega_{0}}~{};~{}B(t)=B_{0}~{}.$ (38)

Therefore, substituting Eqn.(38) in Eqn(s).(35,36), the reduced form of the NC
parameters for this case are as follows,

$\displaystyle\theta(t)=\dfrac{8Me^{-\Gamma\,t}}{q^{2}B_{0}^{2}e^{-2\Gamma\,t}+4M^{2}\omega^{2}_{0}\,}$
$\displaystyle\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma
e^{-2\Gamma\,t}}{4M}+\omega_{0}^{2}\,\left(M\sigma\,-1\right)}-\dfrac{qB_{0}e^{-\Gamma\,t}}{2M}\right]$
$\displaystyle\Omega(t)$
$\displaystyle=-qB_{0}\,+2e^{\Gamma\,t}\sqrt{M\Delta\,-M^{2}\omega_{0}^{2}\,}~{}~{}~{}.$
(39)

It can be checked that in the limit $B\rightarrow 0$, the expressions for
$\theta(t)$ and $\Omega(t)$ reduce to those in [19]. Substituting these
relations in the expression for $c(t)$ in Eqn.(9), we get,

$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{q^{2}B_{0}^{2}e^{-2\Gamma\,t}+4M^{2}\omega_{0}^{2}}\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}e^{-\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma
e^{-2\Gamma\,t}}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$ (40)
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}e^{-\Gamma\,t}-\dfrac{q^{2}B_{0}^{2}}{M}e^{-2\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right]+\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}~{}.$

Substituting the expressions of $a(t)\,,\rho(t)$ and $c(t)$ in Eqn.(28) , we
can get an expression for the phase in a closed form as,

$\displaystyle\small\Theta_{\,n\,,\,l}(t)\,$
$\displaystyle=(n+l)\int_{0}^{t}\left[c(t)-\dfrac{a}{\rho^{2}}\right]~{}dT=(n+l)\left[\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}-\dfrac{\sigma}{\mu^{2}}\right]t$
$\displaystyle+\dfrac{(n+l)\omega_{0}\sqrt{(M\sigma-1)}}{\Gamma}\,log\,\dfrac{\omega_{0}\sqrt{M\sigma-1}\,e^{\Gamma\,t}+\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{2\Gamma\,t}}}{\omega_{0}\sqrt{M\sigma-1}+\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}}$
$\displaystyle+\dfrac{(n+l)\omega_{0}}{\Gamma}\left[tan^{-1}\dfrac{\omega_{0}e^{\Gamma\,t}}{\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{2\Gamma\,t}}}-tan^{-1}\dfrac{\omega_{0}}{\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}}\right]$
$\displaystyle+\dfrac{(n+l)\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{M\Gamma}\left[tanh^{-1}\dfrac{2M\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{2\Gamma\,t}}}{qB_{0}}-tanh^{-1}\dfrac{2M\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}}{qB_{0}}\right]$
$\displaystyle-\dfrac{(n+l)\sqrt{(\Delta-M\omega_{0}^{2})\sigma}}{\Gamma}\left[tanh^{-1}\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma+4M\omega_{0}^{2}(M\sigma-1)e^{2\Gamma\,t}}{q^{2}B_{0}^{2}\sigma}}-tanh^{-1}\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma+4M\omega_{0}^{2}(M\sigma-1)}{q^{2}B_{0}^{2}\sigma}}\right]$
$\displaystyle-\dfrac{(n+l)\omega_{0}}{\Gamma}\left[tan^{-1}\dfrac{2M\omega_{0}e^{\Gamma\,t}}{qB_{0}}-tan^{-1}\dfrac{2M\omega_{0}}{qB_{0}}\right]+\dfrac{(n+l)\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{2\Gamma\,M}\,log\,\dfrac{q^{2}B_{0}^{2}e^{-2\Gamma\,t}+4M^{2}\omega_{0}^{2}}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}~{}.$
(41)

$\langle b\rangle$ Set-I , Case II Here we set the constants

$\displaystyle\vartheta=\Gamma~{},~{}\delta=0~{},\Lambda=\Gamma~{}.$ (42)

So, the situation can be depicted by the following relations,

$\displaystyle f(t)=e^{-\Gamma
t}~{};~{}\omega(t)={\omega_{0}}~{};~{}B(t)=B_{0}\,e^{\Gamma\,t}~{}.$ (43)

Therefore, substituting Eqn.(42) in Eqn(s).(35,36), the reduced form of the NC
parameters for this case are as follows,

$\displaystyle\theta(t)=\dfrac{8Me^{-\Gamma\,t}}{q^{2}B_{0}^{2}+4M^{2}\omega^{2}_{0}\,}\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}\,\left(M\sigma\,-1\right)}-\dfrac{qB_{0}}{2M}\right]~{},~{}\Omega(t)=-qB_{0}\,e^{\Gamma\,t}+2e^{\Gamma\,t}\sqrt{M\Delta\,-M^{2}\omega_{0}^{2}\,}~{}.$
(44)

The point that is to be noted is that the multiplication of the two time
dependent NC parameters obtained for this case reduces to a constant value and
later we observe that the constant value is equal to the same found for
another case discussed in Eqn.(70). It can be checked that in the limit
$B\rightarrow 0$, the expressions for $\theta(t)$ and $\Omega(t)$ reduce to
those obtained in [19]. Substituting these relations in the expression for
$c(t)$ in Eqn.(9), we get,

$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$
(45)
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right]+\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}~{}.$

Substituting the expressions of $a(t)\,,\rho(t)$ and $c(t)$ in Eqn.(28) , we
can get an expression for the phase in a closed form in the following way.

$\displaystyle\Theta_{\,n\,,\,l}(t)\,=\dfrac{(n\,+\,l)}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}$
$\displaystyle\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right]t+(n+l)\left[\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}-\dfrac{\sigma}{\mu^{2}}\right]t~{}.$
(46)

In this case the phase is varying linearly with respect to time. In the limit
$B_{0}\rightarrow\,0$ , we can easily recover the same form of the phase
factor corresponding to the solution set Ib obtained in [19].
$\langle c\rangle$ Set-I , Case III  Here we set the constants

$\displaystyle\vartheta=\Gamma~{},~{}\delta=0~{},~{}\Lambda=-\Gamma~{}.$ (47)

So, the situation can be depicted by the following relations,

$\displaystyle
f(t)=e^{-\Gamma\,t}~{},~{}\omega(t)={\omega_{0}}~{},~{}B(t)=B_{0}\,e^{-\Gamma\,t}~{}.$
(48)

Therefore, substituting Eqn.(47) in Eqn(s).(35,36), the reduced form of the NC
parameters for this case are as follows,

$\displaystyle\theta(t)=\dfrac{8Me^{-\Gamma\,t}}{q^{2}B_{0}^{2}e^{-4\Gamma\,t}+4M^{2}\omega^{2}_{0}}$
$\displaystyle\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma\,e^{-4\Gamma\,t}}{4M}+\omega_{0}^{2}(M\sigma-1)}-\dfrac{qB_{0}e^{-2\Gamma\,t}}{2M}\right]$
$\displaystyle\Omega(t)$
$\displaystyle=-qB_{0}e^{-\Gamma\,t}+2e^{\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}~{}.$
(49)

Substituting these relations in the expression for $c(t)$ in Eqn.(9), we get,

$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{q^{2}B_{0}^{2}e^{-4\Gamma\,t}+4M^{2}\omega_{0}^{2}}\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}e^{-2\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma
e^{-4\Gamma\,t}}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$ (50)
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}e^{-2\Gamma\,t}-\dfrac{q^{2}B_{0}^{2}e^{-4\Gamma\,t}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right]+\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}~{}.$

Substituting the expressions of $a(t)\,,\rho(t)$ and $c(t)$ in Eqn.(28) , we
can get an expression for the phase factor in a closed form as ,

$\displaystyle\small\Theta_{\,n\,,\,l}(t)\,$
$\displaystyle=(n+l)\int_{0}^{t}\left[c(t)-\dfrac{a}{\rho^{2}}\right]~{}dT=(n+l)\left[\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}-\dfrac{\sigma}{\mu^{2}}\right]t$
$\displaystyle+\dfrac{(n+l)\omega_{0}\sqrt{(M\sigma-1)}}{2\Gamma}\,log\,\dfrac{\omega_{0}\sqrt{M\sigma-1}\,e^{2\Gamma\,t}+\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{4\Gamma\,t}}}{\omega_{0}\sqrt{M\sigma-1}+\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}}$
$\displaystyle+\dfrac{(n+l)\omega_{0}}{2\Gamma}\left[tan^{-1}\dfrac{\omega_{0}e^{2\Gamma\,t}}{\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{4\Gamma\,t}}}-tan^{-1}\dfrac{\omega_{0}}{\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}}\right]$
$\displaystyle+\dfrac{(n+l)\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{2M\Gamma}\left[tanh^{-1}\dfrac{2M\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{4\Gamma\,t}}}{qB_{0}}-tanh^{-1}\dfrac{2M\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}}{qB_{0}}\right]$
$\displaystyle-\dfrac{(n+l)\sqrt{(\Delta-M\omega_{0}^{2})\sigma}}{2\Gamma}\left[tanh^{-1}\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma+4M\omega_{0}^{2}(M\sigma-1)e^{4\Gamma\,t}}{q^{2}B_{0}^{2}\sigma}}-tanh^{-1}\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma+4M\omega_{0}^{2}(M\sigma-1)}{q^{2}B_{0}^{2}\sigma}}\right]$
$\displaystyle-\dfrac{(n+l)\omega_{0}}{2\Gamma}\left[tan^{-1}\dfrac{2M\omega_{0}e^{2\Gamma\,t}}{qB_{0}}-tan^{-1}\dfrac{2M\omega_{0}}{qB_{0}}\right]+\dfrac{(n+l)\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{4\Gamma\,M}\,log\,\dfrac{q^{2}B_{0}^{2}e^{-4\Gamma\,t}+4M^{2}\omega_{0}^{2}}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}~{}.$
(51)

$\langle d\rangle$ Set-I , Case IV  Here we set the constants

$\displaystyle\vartheta=\delta=\Lambda=\Gamma~{}.~{}$ (52)

So, the situation can be depicted by the following relations,

$\displaystyle
f(t)=e^{-\Gamma\,t}~{};~{}\omega(t)={\omega_{0}}e^{-\Gamma\,t/2}~{};~{}B(t)=B_{0}\,e^{\Gamma\,t}~{}.$
(53)

Substituting Eqn.(52) in Eqn(s).(35,36), the reduced form of the NC parameters
for this case are as follows,

$\displaystyle\theta(t)=\dfrac{8Me^{-\Gamma\,t}}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}e^{-\Gamma\,t}(M\sigma-1)}-\dfrac{qB_{0}}{2M}\right]\,,\,\Omega(t)=-qB_{0}e^{\Gamma\,t}+2e^{\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}~{}.$
(54)

Substituting these relations in the expression for $c(t)$ in Eqn.(9), we get,

$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\left[\left(4M^{2}\omega_{0}^{2}e^{-\Gamma\,t}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}e^{-\Gamma\,t}(M\sigma-1)}\right.$
(55)
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}e^{-\Gamma\,t}-\dfrac{q^{2}B_{0}^{2}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\right]+\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}e^{-\Gamma\,t}}~{}.$

We are able to obtain the exact form of the phase factor and it has been shown
in the Appendix.

###  4.2 Solution Set-II for Ermakov-Pinney equation: Rationally decaying
solutions

####  4.2.1 The Solution Set

We now consider rationally decaying solutions of the EP equation similar to
that used in [27] which is of the form,

$\displaystyle
a(t)=\dfrac{\sigma\,\left(1+\dfrac{2}{k}\right)^{\,(k+2)/k}}{(\Gamma{t}+\chi)^{\,(k+2)/k}}~{},~{}b(t)=\dfrac{\Delta\,\left(\dfrac{k}{k+2}\right)^{(2-k)/k}}{(\Gamma{t}+\chi)^{\,(k-2)/k}}~{},~{}\rho(t)=\dfrac{\mu\left(1+\dfrac{2}{k}\right)^{1/k}}{(\Gamma{t}+\chi)^{1/k}}~{};$
(56)

where $\sigma$, $\Delta$, $\mu$, $\Gamma$ and $\chi$ are constants such that
$(\Gamma{t}+\chi)~{}\neq~{}0$, and $k$ is an integer. Substituting the
expressions of $a(t)$, $b(t)$, and $\rho(t)$ in the EP equation, we can easily
verify the relation between these constants to be as follows,

$\Gamma^{2}\mu=(k+2)^{2}\,(\sigma\Delta\mu-\frac{\xi^{2}\sigma^{2}}{\mu^{3}}).$
(57)

####  4.2.2 Study of the corresponding eigenfunctions

The eigenfunction of the invariant operator $I(t)$ [given by Eqn.(24)] for
this solution Set-II is of the following form,

$\displaystyle\phi_{n\,,\,m-n}(r,\theta)=\lambda_{n}\,\dfrac{{(i\mu)}^{\,m}}{\sqrt{m!}}\left[\dfrac{k+2}{k(\Gamma{t}+\chi)}\right]^{m/k}r^{n-m}e^{i(m-n)\theta-\dfrac{[\sigma\,(k+2)\,+\,i\mu^{2}\Gamma]\,\,(\Gamma{t}+\chi)^{2/k}\,\,\,k^{2/k}}{2\sigma\,(k+2)^{\,(k+2)/k}\mu^{2}}r^{2}}$
$\displaystyle\times\,\,\,U\left(-m,1-m+n,\,\dfrac{r^{2}[k(\Gamma{t}+\chi)]^{2/k}}{\mu^{2}\left(k+2\right)^{2/k}}\,\right)$
(58)

where $\lambda_{n}$ is given by

$\displaystyle\lambda_{n}^{\,2}=\dfrac{1}{\pi\,n!\mu^{2n+2}}\left[\dfrac{k(\Gamma{t}+\chi)}{k+2}\right]^{2(1+n)/k}.$
(59)

In order to get the eigenfunction of the Hamiltonian $H(t)$, we need to
calculate the associated phase factor. Once again for this we need to fix up
the forms of the damping factor $f(t)$, angular frequency $\omega(t)$ of the
oscillator and the applied magnetic field $B(t)$. In order to explore the
solution of $H(t)$ for rationally decaying coefficients, we choose a
rationally decaying form for $\omega(t)$, $B(t)$ and set $f(t)=1$. Thus, we
have the following relations,

$\displaystyle\eta(t)=0\,\,\Rightarrow\,\,f(t)=1~{},~{}\omega(t)=\dfrac{\omega_{0}}{(\Gamma\,t+\chi)}~{},~{}B(t)=\dfrac{B_{0}}{(\Gamma\,t+\chi)}.$
(60)

$\langle e\rangle$ Set-II , Case I

As we want to study how the nature of the rationally decaying solution gets
altered when the system is placed in a magnetic field, we set $k=2$ in
Eqn.(56). The system has been studied without applying any external field for
this particular $k$ parameter in an earlier communication [19].

When we set $k=2$,the set $a(t)$, $b(t)$ and $\rho(t)$ takes the following
simplified form,

$\displaystyle
a(t)=\dfrac{4\sigma}{(\Gamma{t}+\chi)^{\,2}}\,\,,\,\,b(t)\,=\,\Delta\,\,,\,\,\rho(t)=\left[\dfrac{2\mu^{\,2}}{\Gamma{t}+\chi}\right]^{1/2}.$
(61)

Substituting the expressions for $a(t)$, $b(t)$, $\omega(t)$ , $f(t)$ and
$B(t)$ in the Eqn(s).(7, 8), we get the time dependent NC parameters as,

$\displaystyle\theta(t)=\dfrac{8M}{q^{2}B_{0}^{2}+4M^{2}\omega^{2}_{0}}$
$\displaystyle\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{M}+\omega_{0}^{2}4\sigma
M-\omega_{0}^{2}(\Gamma\,t+\chi)^{2}}-\dfrac{qB_{0}}{2M}(\Gamma\,t+\chi)\right]$
$\displaystyle\Omega(t)=-\dfrac{qB_{0}}{(\Gamma\,t+\chi)}+2\sqrt{M\Delta-\dfrac{M^{2}\omega_{0}^{2}}{(\Gamma\,t+\chi)^{2}}}~{}.$
(62)

Substituting these relations in the expression for $c(t)$ in Eqn.(9) gives,

$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{4M^{2}\omega^{2}_{0}+q^{2}B_{0}^{2}}\left[\left(\dfrac{4M^{2}\omega_{0}^{2}}{(\Gamma\,t+\chi)^{2}}+\dfrac{2qB_{0}\sqrt{M\Delta(\Gamma\,t+\chi)^{2}-M^{2}\omega_{0}^{2}}}{(\Gamma\,t+\chi)^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{M}-\omega_{0}^{2}(\Gamma\,t+\chi)^{2}+4\omega_{0}^{2}\sigma
M}\right.$ (63)
$\displaystyle\left.-\dfrac{2qB_{0}M\omega_{0}^{2}}{\Gamma\,t+\chi}-\dfrac{q^{2}B_{0}^{2}\sqrt{M\Delta(\Gamma\,t+\chi)^{2}-M^{2}\omega_{0}^{2}}}{M(\Gamma\,t+\chi)}\right]+\sqrt{\dfrac{\Delta}{M}-\dfrac{\omega_{0}^{2}}{(\Gamma\,t+\chi)^{2}}}~{}.$

The additional terms that appear in the expression due to the presence of the
magnetic field are mostly seen to be decaying functions of time. Their
contribution becomes more evident when we study the evolution of expectation
value of energy with time in a later section. We are able to obtain the exact
form of the phase factor and it has been shown in the Appendix.

$\langle f\rangle$ Set-II , Case II

It is observed from the solution set given by Eqn.(56), that the time
dependent parameters $a(t)$ and $\rho(t)$ vanish, while $b(t)$ diverges if we
set $k=-2$. In order to avoid this and study the system at this critical value
of $k$, we choose the corresponding solution set to be,

$\displaystyle
a(t)=\dfrac{\sigma}{(\Gamma{t}+\chi)^{\,(k+2)/k}}~{},~{}b(t)=\dfrac{\Delta}{(\Gamma{t}+\chi)^{\,(k-2)/k}}~{},~{}\rho(t)=\dfrac{\mu}{(\Gamma{t}+\chi)^{1/k}}.$
(64)

We now set $k=-2$ in Eqn. (64) to obtain the following set,

$\displaystyle
a=\sigma~{}~{},~{}~{}b(t)=\dfrac{\Delta}{(\Gamma\,t+\chi)^{2}}~{}~{},~{}~{}\rho=\mu\sqrt{\Gamma\,t+\chi}\,.$
(65)

Substituting these relations in the EP equation the constraint condition is
found to be,

$-\mu^{4}\Gamma^{2}+4\sigma\Delta\mu^{4}=4\,\xi^{2}\sigma^{2}\,.$ (66)

The above relation matches with that found in Eqn.(31) while considering
$\vartheta=\Gamma$.

As the Eqn.(58) vanish at $k=-2$, the eigenfunction of the invariant for this
solution is needed to be calculated separately and it is of the following
form,

$\displaystyle\phi_{n\,,\,m-n}(r,\theta)=\lambda_{n}\,\dfrac{{(i\mu\sqrt{(\Gamma\,t+\chi})}^{\,m}}{\sqrt{m!}}r^{n-m}e^{i(m-n)\theta-\dfrac{2\sigma-i\mu^{2}\Gamma}{4\sigma\mu^{2}(\Gamma\,t+\chi)}r^{2}}~{}U\left(-m,1-m+n,\,\dfrac{r^{2}}{\mu^{2}(\Gamma\,t+\chi)}\,\right)$
(67)

where $\lambda_{n}$ is given by

$\displaystyle\lambda_{n}^{\,2}=\dfrac{1}{\pi\,n![\mu^{2}(\Gamma\,t+\chi)]^{1+n}}.$
(68)

Again we consider the same explicit forms of the damping factor $f(t)$,
angular frequency $\omega(t)$ and applied magnetic field $B(t)$ as in
Eqn.(60),

$\displaystyle
f(t)=1~{},~{}\omega(t)=\dfrac{\omega_{0}}{(\Gamma\,t+\chi)}~{},~{}B(t)=\dfrac{B_{0}}{(\Gamma\,t+\chi)}~{}.$
(69)

Substituting the expressions for $a(t)$, $b(t)$, $\omega(t)$ , $f(t)$ and
$B(t)$ in the Eqn(s).(7, 8), we get the time dependent NC parameters as,

$\displaystyle\theta(t)=\dfrac{8M(\Gamma\,t+\chi)}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}-\dfrac{qB_{0}}{2M}\right]~{},~{}\Omega(t)=2\dfrac{\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{(\Gamma\,t+\chi)}-\dfrac{qB_{0}}{(\Gamma\,t+\chi)}~{}.$
(70)

Once again it is interesting to note that a constant value is found after
multiplication of the two time dependent NC parameters obtained above. Here we
recall Eqn.(44) where we discussed that the multiplication of two time
dependent NC parameters reduces to a constant value which is equal to the same
obtained for this case.

Substituting these relations in the expression for $c(t)$ in Eqn.(9) gives,

$\displaystyle c(t)$ $\displaystyle=$
$\displaystyle\dfrac{1}{(4M^{2}\omega^{2}_{0}+q^{2}B_{0}^{2})(\Gamma\,t+\chi)}\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$
(71)
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{M}\right]+\dfrac{1}{(\Gamma\,t+\chi)}\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}~{}.$

Substituting these expressions for $a(t)$, $\rho(t)$ and $c(t)$ in Eqn.(28),
we get the following expression for the phase factor in a closed form as,

$\displaystyle\Theta_{\,n,l\,}(t)$ $\displaystyle=$
$\displaystyle\dfrac{(n+l)}{(4M^{2}\omega^{2}_{0}+q^{2}B_{0}^{2})\Gamma}\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{M}\right]ln\dfrac{(\Gamma\,t+\chi)}{\chi}+\dfrac{(n+l)}{\Gamma}\left[\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}-\dfrac{\sigma}{\mu^{2}}\right]~{}ln\dfrac{(\Gamma\,t+\chi)}{\chi}~{}.$

##  5 Analysis of the expectation value of energy

In this section, we intend to calculate the expectation value of energy. It is
shown in [19] that 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 expressed
as,

$\displaystyle\braket{E_{n,m-n}(t)}$
$\displaystyle=\dfrac{1}{2}\,(n+m+1)\left[b(t)\rho^{2}(t)+\dfrac{a(t)}{\rho^{2}(t)}+\dfrac{\dot{\rho}^{2}(t)}{a(t)}\right]+c(t)\,(n-m)\,\,.$
(73)

Substituting the expression of $c(t)$ in the above equation, the expectation
value of energy for our model takes the following form,

$\displaystyle\braket{E_{n,m-n}(t)}$
$\displaystyle=\dfrac{1}{2}\,(n+m+1)\left[b(t)\rho^{2}(t)+\dfrac{a(t)}{\rho^{2}(t)}+\dfrac{\dot{\rho}^{2}(t)}{a(t)}\right]$
$\displaystyle+\dfrac{(n-m)}{2}\left[\dfrac{qB(t)f(t)}{M}\left(1+\dfrac{\theta(t)\Omega(t)}{4}\right)+\dfrac{\Omega(t)f(t)}{M}+\left(\dfrac{q^{2}B^{2}(t)f(t)}{4M}+\dfrac{M\omega^{2}(t)}{f(t)}\right)\theta(t)\right]~{};$
(74)

which reduces to the same obtained in [19] in the limit $B\rightarrow\,0$.
The energy expression depends on the charge explicitly. It contains both terms
having linear and quadratic dependence on charge. So the energy does not
remain invariant when the charge of the particle changes its sign. Another
notable point is that even when the frequency of oscillation
$\omega{\rightarrow}0$, and the applied field $B\rightarrow\,0$; the
expectation value of energy is non-zero. This is because all the three
parameters of the Hamiltonian $a(t)$, $b(t)$ and $c(t)$ are finite even as
$\omega{\rightarrow}0,B{\rightarrow}0$, as is clear from the Eqn(s).(7,8,9).
Now we will proceed to study the time-dependent behaviour of
$\braket{E_{n,m-n}(t)}$ for various types of damping and applied magnetic
field.

###  5.1 Solution Set-I: Exponentially decaying solution

For the exponentially decaying solution given by Eqn.(30), the energy
expectation value takes the following form,

$\braket{E_{n,m-n}(t)}=(n+m+1)\mu^{2}\Delta+c(t)\,(n-m)$ (75)

where we have set the constant $\xi^{2}$ to unity and used the constraint
relation given by Eqn.(31).

$\langle A\rangle$ Set-I , Case I Here we set $f(t)=e^{-\Gamma\,t}$ ,
$\omega(t)=\omega_{0}$ and $B(t)=B_{0}$. With this the energy expression for
the ground state takes the form,

$\displaystyle\braket{E_{n,-n}(t)}$
$\displaystyle=(n+1)\mu^{2}\Delta+\dfrac{n}{q^{2}B_{0}^{2}e^{-2\Gamma\,t}+4M^{2}\omega_{0}^{2}}\left[-2qB_{0}M\omega_{0}^{2}e^{-\Gamma\,t}-\dfrac{q^{2}B_{0}^{2}}{M}e^{-2\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right.$
$\displaystyle\left.+\left(4M^{2}\omega_{0}^{2}+2qB_{0}e^{-\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma
e^{-2\Gamma\,t}}{4M}+\omega_{0}^{2}(M\sigma-1)}\right]+n\,\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}.$
(76)

The nature of this energy expectation value depends on the value of the
constants. Specifically, the sign of the charge plays a crucial role for
determining the nature of this expectation value. It is noteworthy that in
both the limits $t\rightarrow\infty$ and $B\rightarrow\,0$ this energy
expression gains the same constant value which is already found in [19] for a
damped oscillator in time dependent NC space. Apart from it, an inclusion of
constant magnetic field in the system considered in [19] also makes the
Hamiltonian non-hermitian after a certain limit of time beyond which the
energy becomes imaginary. The condition for getting the expectation value of
energy to be real is as follows,

$t~{}\leq~{}\dfrac{1}{2\Gamma}\,ln\dfrac{q^{2}B_{0}^{2}\sigma}{4M\omega_{0}^{2}(1-M\sigma)}~{}.$
(77)

It is interesting to note that the upper bound of time below which the energy
expectation value remains real does not depend on the sign of charge of the
oscillator, although the value of energy itself does. The nature of variation
of energy expectation value with time is shown in Fig.1. Though the energy
shows an initial decrease, eventually it tends to be a constant over time.

$\langle B\rangle$ Set-I , Case II
Here we set $f(t)=e^{-\Gamma\,t}$ , $\omega(t)=\omega_{0}$ and
$B(t)=B_{0}\,e^{\Gamma\,t}$. With this the energy expression for the ground
state takes the form,

$\displaystyle\braket{E_{n,-n}(t)}$
$\displaystyle=(n+1)\mu^{2}\Delta+\dfrac{n}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}\left[\left(4M^{2}\omega_{0}^{2}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}\right.$
$\displaystyle\left.-2qB_{0}M\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right]+n\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}=constant~{}.$
(78)

As expected we observe from Fig. 1, the energy is a constant over time. In the
limit $B\rightarrow\,0$, the constant value of energy reduces to the same
obtained in [19] for a damped oscillator in time dependent NC space. $\langle
C\rangle$ Set-I , Case III
Here we set $f(t)=e^{-\Gamma\,t}$ , $\omega(t)=\omega_{0}$ and
$B(t)=B_{0}e^{-\Gamma\,t}$. With this the energy expression for the ground
state takes the form,

$\displaystyle\braket{E_{n,-n}(t)}$
$\displaystyle=(n+1)\mu^{2}\Delta+\dfrac{n}{q^{2}B_{0}^{2}e^{-4\Gamma\,t}+4M^{2}\omega_{0}^{2}}\left[-2qB_{0}M\omega_{0}^{2}e^{-2\Gamma\,t}-\dfrac{q^{2}B_{0}^{2}e^{-4\Gamma\,t}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right.$
$\displaystyle\left.+\left(4M^{2}\omega_{0}^{2}+2qB_{0}e^{-2\Gamma\,t}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma
e^{-4\Gamma\,t}}{4M}+\omega_{0}^{2}(M\sigma-1)}\right]+n~{}\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}~{}.$
(79)

The nature of variation of the energy expectation value with time has almost
the same characteristics as obtained in Case I. This is also observed
graphically from Fig. 1. However, a closer observation tells us that in Case
III the rate of decay is faster than in Case I for the same set of parameters.
It must be because in Case III, unlike in Case I, the applied field is
decaying as well with respect to time. The upper bound of time beyond which
the system becomes non-physical (since the energy ceases to be real after this
time) is also half of that obtained in Eqn.(77). Here the bound is found to be

$t\leq\,\dfrac{1}{4\Gamma}ln\dfrac{q^{2}B_{0}^{2}\sigma}{4M\omega_{0}^{2}(1-M\sigma)}~{}.$
(80)

An important inference that can be drawn from the above study is that, in the
presence of the external magnetic field, the energy of a damped oscillator
decays with time if the field either decays with time or is atleast a
constant. If the field tends to grow as fast as the damping factor decays,
then the energy of the oscillator tends to be a constant with time. In Fig. 1
we have also plotted the evolution of energy with time in this situation if
the applied field is turned off. We see that if B=0, then the energy is a
constant over time. This is expected from the energy expectation value
expressions given by Eqns. [76, 78 and 79]. Even a constant magnetic field is
able to bring about time variation in this energy value.

Figure 1: A study of the variation of expectation value of energy, scaled by
$\frac{1}{\omega_{0}}$ ($\frac{\langle E\rangle}{\omega_{0}}$) in order to
make it dimensionless, as we vary $\Gamma$t (again a dimensionless quantity).
Here we consider mass M=1, charge q=1, magnetic field $B_{0}$=$10^{2}$,
$\mu$=1,$\Delta$=$10^{7}$, $\sigma$=$10^{7}$, $\omega_{0}$=$10^{3}$ and
$\Gamma$=1 in natural units. The constants n=1 and m=0. The expectation value
of energy $\langle E\rangle$ is calculated for exponentially decaying solution
set when Case I: $f(t)=e^{-\Gamma\,t}$ , $\omega(t)=\omega_{0}$ and
$B(t)=B_{0}$; Case II: $f(t)=e^{-\Gamma\,t}$ , $\omega(t)=\omega_{0}$ and
$B(t)=B_{0}\,e^{\Gamma\,t}$; Case III: $f(t)=e^{-\Gamma\,t}$ ,
$\omega(t)=\omega_{0}$ and $B(t)=B_{0}e^{-\Gamma\,t}$ and Case IV:
$f(t)=e^{-\Gamma\,t}$ , $\omega(t)=\omega_{0}e^{-\Gamma\,t/2}$ and
$B(t)=B_{0}e^{\Gamma\,t}$. While for Case I and Case III the energy first
decreases, then becomes constant with time, for Case II the energy remains
constant as we vary time. For Case IV the behaviour of energy with time is
seen to be extremely interesting. It first decreases and then increases with
time. Along with these, we have also plotted what happens in the absence of
magnetic field for comparison. When the angular frequency of oscillation is a
constant (Case I, Case II and Case III), if the magnetic field is set to zero,
then the energy of the oscillator is a constant with time. So, the magnetic
field, even when it is a constant brings about time variation in energy for an
exponentially damped oscillator having a constant frequency. However, when the
angular frequency is decaying exponentially too (Case IV), then even when B=0,
the energy decays with time. Nevertheless, the variation of energy is
remarkably different from that seen in the presence of the field for Case IV.

$\langle D\rangle$ Set-I, Case IV
Here we set $f(t)=e^{-\Gamma\,t}$ , $\omega(t)=\omega_{0}e^{-\Gamma\,t/2}$ and
$B(t)=B_{0}e^{\Gamma\,t}$. With this the energy expression for the ground
state takes the form,

$\displaystyle\braket{E_{n,-n}(t)}$
$\displaystyle=(n+1)\mu^{2}\Delta+\dfrac{n}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\left[-2qB_{0}M\omega_{0}^{2}e^{-\Gamma\,t}-\dfrac{q^{2}B_{0}^{2}}{M}\sqrt{M\Delta-M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\right.$
$\displaystyle\left.+\left(4M^{2}\omega_{0}^{2}e^{-\Gamma\,t}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}e^{-\Gamma\,t}(M\sigma-1)}\right]+n\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}e^{-\Gamma\,t}}~{}.$
(81)

It can be verified that the above energy expectation value reduces to a
decaying form as obtained in [19] in the absence of magnetic field. However,
it is interesting to note that for certain choice of parameters (as shown in
Fig. 1) the energy may also exhibit an initial growth with time. Infact for
the given set of parameters in Fig. 1, it initially decays and then increases.
Eventually, as $t~{}\rightarrow~{}\infty$, the energy tends to be a constant.
In this situation, if we turn off the applied magnetic field, then we see from
Fig 1, the energy decays off with time. However, due to the absence of the
exponentially growing field, the energy of the oscillator is seen to be much
lower. Also the exponential growth in energy seen in the presence of the field
is absent within the time range studied. The system also possess two lower
bounds of time below which it becomes non-physical due to the imaginary value
of energy. The conditions are as follows,

$\displaystyle
t~{}\geq~{}\dfrac{1}{\Gamma}ln\left[\dfrac{M\omega_{0}^{2}}{\Delta}\right]~{};~{}t~{}\geq~{}\dfrac{1}{\Gamma}ln\left[\dfrac{4M\omega_{0}^{2}(1-M\sigma)}{q^{2}B_{0}^{2}\sigma}\right]~{}.$
(82)

The greater of the two bounds serves as the actual lower bound.

###  5.2 Solution Set-II: Rationally decaying solution

In previous section we considered two different solution set generated from
Eqn.(56). The special form of the solution shown in Eqn.(61) is directly
produced by substituting $k=2$ in the Eqn.(56) and the special form of the
solution shown in Eqn.(65) is obtained by substituting $k=-2$ in the modified
form of Eqn.(56).

####  5.2.1 Set-II, Case I

For the rationally decaying solution given by Eqn.(61), the energy expectation
value takes the following form

$\braket{E_{n,m-n}(t)}=\dfrac{(n+m+1)}{2(\Gamma\,t+\chi)}\left[2\left(\dfrac{\sigma}{\mu^{2}}+\Delta\mu^{2}\right)+\dfrac{\mu^{2}\Gamma^{2}}{8\sigma}\right]+(n-m)c(t)~{}.$
(83)

where we have set the constant $\xi^{2}$ to unity and used the constraint
relation given by Eqn.(57). Here we set $f(t)=1$ ,
$\omega(t)=\omega_{0}/(\Gamma\,t+\chi)$ and $B(t)=B_{0}/(\Gamma\,t+\chi)$.
With this the energy expression for the ground state takes the form,

$\displaystyle\braket{E_{n,-n}(t)}=\dfrac{(n+1)}{2(\Gamma\,t+\chi)}\left[2\left(\dfrac{\sigma}{\mu^{2}}+\Delta\mu^{2}\right)+\dfrac{\mu^{2}\Gamma^{2}}{8\sigma}\right]+\dfrac{n}{4M^{2}\omega^{2}_{0}+q^{2}B_{0}^{2}}\left[-\dfrac{2qB_{0}M\omega_{0}^{2}}{\Gamma\,t+\chi}-\dfrac{q^{2}B_{0}^{2}\sqrt{M\Delta(\Gamma\,t+\chi)^{2}-M^{2}\omega_{0}^{2}}}{M(\Gamma\,t+\chi)}\right.$
$\displaystyle\left.+\left(\dfrac{4M^{2}\omega_{0}^{2}}{(\Gamma\,t+\chi)^{2}}+\dfrac{2qB_{0}\sqrt{M\Delta(\Gamma\,t+\chi)^{2}-M^{2}\omega_{0}^{2}}}{(\Gamma\,t+\chi)^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{M}-\omega_{0}^{2}(\Gamma\,t+\chi)^{2}+4\omega_{0}^{2}\sigma
M}\right]+n\sqrt{\dfrac{\Delta}{M}-\dfrac{\omega_{0}^{2}}{(\Gamma\,t+\chi)^{2}}}~{};$
(84)

which is a decaying function of time and is seen to reduce to the same
obtained in [19] in the limit $B\rightarrow\,0$. The time range beyond which
the system becomes non-physical due to imaginary energy expectation value is
as follows,

$\dfrac{1}{\Gamma}\left(\omega_{0}\sqrt{\dfrac{M}{\Delta}}-\chi\right)\leq~{}t\leq~{}\dfrac{1}{\Gamma}\left[\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{M\omega_{0}^{2}}+4\sigma
M}-\chi\right]~{}.$ (85)

In Fig. 2 we have done a comparative study of the time variation of energy in
this situation, when the applied magnetic field is present and when it is
turned off. We see from the Figure that in both these cases the energy decays
with time. However the energy of the oscillator is higher in the presence of
the field due to the presence of the magnetic energy in the system.

Figure 2: A study of the variation of expectation value of energy, scaled by
$\frac{1}{\omega_{0}}$ ($\frac{\langle E\rangle}{\omega_{0}}$) in order to
make it dimensionless, as we vary $\Gamma$t (again a dimensionless quantity).
Here we consider mass M=1, charge q=1, magnetic field $B_{0}$=$10^{20}$,
$\mu$=1,$\Delta$=$10^{7}$, $\sigma$=$10^{7}$, $\omega_{0}$=$10^{3}$ and
$\Gamma$=1 in natural units. The constants n=1 and m=0. The expectation value
of energy $\langle E\rangle$ is calculated for rationally decaying solution
set when $\langle A\rangle$ Case I:
$a(t)=\dfrac{4\sigma}{(\Gamma{t}+\chi)^{\,2}}\,\,,\,\,b(t)\,=\,\Delta\,\,,\,\,\rho(t)=\left[\dfrac{2\mu^{\,2}}{\Gamma{t}+\chi}\right]^{1/2}$
and $\langle B\rangle$ Case II:
$a=\sigma~{}~{},~{}~{}b(t)=\dfrac{\Delta}{(\Gamma\,t+\chi)^{2}}~{}~{},~{}~{}\rho=\mu\sqrt{\Gamma\,t+\chi}$.
For both Case I and Case II the energy decays with time. However, the rate of
decay is higher for Case I as compared to Case II. This is expected since two
of the Hamiltonian parameters $a(t)$ and $\rho(t)$ are decaying with time for
Case I, whereas only one parameter $b(t)$ is decaying while another one
$\rho(t)$ is increasing with time for Case II. For comparison, we have also
plotted Case I and II, in the absence of the magnetic field in the same plot.
It is noteworthy, that although the energy of the oscillator decays even in
the absence of the field, when we apply the field the energy is enhanced. This
is due to the presence of magnetic energy in the system in this situation.

####  5.2.2 Set-II, Case II

For the rationally decaying solution given by Eqn.(65), the energy expectation
value takes the following form,

$\braket{E_{n,m-n}(t)}=\dfrac{(m+n+1)\mu^{2}\Delta}{\Gamma\,t+\chi}+c(t)(n-m)~{}.$
(86)

where we have set the constant $\xi^{2}$ to unity and used the constraint
relation given by Eqn.(66). Here we set $f(t)=1$ ,
$\omega(t)=\omega_{0}/(\Gamma\,t+\chi)$ and $B(t)=B_{0}/(\Gamma\,t+\chi)$.
With this the energy expression for the ground state takes the form,

$\displaystyle\braket{E_{n,-n}(t)}=\dfrac{(n+1)\mu^{2}\Delta}{\Gamma\,t+\chi}+\dfrac{n}{(4M^{2}\omega^{2}_{0}+q^{2}B_{0}^{2})(\Gamma\,t+\chi)}\left[-2qB_{0}M\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}}{M}\right.$
$\displaystyle\left.+\left(4M^{2}\omega_{0}^{2}+2qB_{0}\sqrt{M\Delta-M^{2}\omega_{0}^{2}}\right)\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}\right]+\dfrac{n}{(\Gamma\,t+\chi)}\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}~{}.$
(87)

As we see from Fig. 2 the corresponding energy expectation value decays off
with time. Infact, it approaches zero at large time. However, the rate of
decay is lower than Case I. This is expected since in Case II, unlike in Case
I, while one of the Hamiltonian parameters $b(t)$ is decreasing with time,
another parameter $\rho(t)$ is growing with time. The resultant energy is
decreasing with time since the rate at which $b(t)$ is decaying (rate
$\sim~{}t^{-2}$) is higher than the rate at which $\rho(t)$ is growing (rate
$\sim~{}t^{1/2}$). When we turn off the field in this situation, we see from
Fig. 2, the energy still decays off due to the decaying angular frequency of
the oscillator. But in this situation the energy of the oscillator is reduced
due to the absence of magnetic energy in the system.

##  6 Conclusion

In conclusion, our primary objective through this study has been to
investigate the effect that an external time-dependent magnetic field has on a
two dimensional damped harmonic oscillator in noncommutative space. The
behaviour of the system under the influence of this time varying field is seen
to be dependent on the nature of the field. For this purpose we have first set
up the Hamiltonian for our system in the presence of a general magnetic field
in noncommutative space. Then we map this Hamiltonian in terms of commutative
variables by using a shift of variables connecting the noncommutative and
commutative space, known in the literature as Bopp-shift. We have then
obtained the exact solution of this time dependent system in the presence of
an applied time varying magnetic field by using the well known Lewis invariant
which in turn leads to a non-linear differential equation known as the
Ermakov-Pinney equation. Then we make various choices of the parameters of the
system and study the solutions depending on these choices as we tune the
applied magnetic field. In this study we have primarily considered two
different sets of parameters for our damped system, namely, exponentially
decaying solutions and rationally decaying solutions. Interestingly, the
solutions obtained make it possible to integrate the expression of the phase
factor exactly thereby giving an exact solution for the eigenstates of the
Hamiltonian. Then we compute the expectation value of the Hamiltonian.
Expectedly, the expectation value of the energy varies with time. For the
exponentially decaying system, the nature of the time dependent magnetic field
crucially determines the nature of evolution of the energy with time. There is
basically an interplay between the damping factor, applied time varying
magnetic field and time dependent angular frequency of the harmonic oscillator
in determining the time evolution. While an exponentially growing magnetic
field is able to maintain the energy to a constant value inspite of damping, a
constant or an exponentially decaying field makes the energy fall off faster
with damping. Remarkably, the presence of an exponentially decaying frequency
along with the damping factor makes the behaviour of the system under the
influence of an exponentially growing field even more interesting. While
initially the energy decays off with time due to the damping present in the
system, later the energy starts growing under the influence of the growing
field. For the rationally decaying situation, even when damping is not
present, a rationally decaying magnetic field in combination with a rationally
decaying angular frequency is able to eventually damp out the energy of the
oscillator. While the decaying oscillation corresponding to the Case I of
rational EP solution cannot remain physical at the zero value of energy due to
the existance of upper bound of time, the same corresponding to the Case II of
rational EP solution is physically damped out to be zero with time. However,
we observe that at a given instant of time, the expectation value of energy is
greater in the presence of the magnetic field than when the field is turned
off. This is because magnetic energy is absent in the system in the latter
case.

## Appendix: Explicit forms of some phases

We discussed about the eigenfunctions corresponding to the exponentially
decaying EP solution set in section $4.1$. We mentioned that the exact form of
the phase factor to construct the eigenfunction of a damped oscillator having
exponentially decaying frequency in the presence of a magnetic field
increasing exponentially with respect to time in NC space can be found [Set I,
Case IV]. The phase factor is as follows,

$\displaystyle\Theta_{\,n\,,\,l}(t)\,=\dfrac{2(n+l)}{\Gamma}\left[\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}}-\sqrt{\dfrac{\Delta}{M}-\omega_{0}^{2}e^{-\Gamma\,t}}\right]+\dfrac{2(n+l)}{\Gamma}\left\\{\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)}-\sqrt{\dfrac{q^{2}B_{0}^{2}\sigma}{4M}+\omega_{0}^{2}(M\sigma-1)e^{-\Gamma\,t}}\right\\}$
$\displaystyle+(n+l)\left[-\dfrac{\sigma}{\mu^{2}}\right]t+\dfrac{i(n+l)}{M\Gamma}\sqrt{q^{2}B_{0}^{2}+4M\Delta}\left[tan^{-1}\dfrac{\sqrt{q^{2}B_{0}^{2}+4M\Delta}}{2\sqrt{M^{2}\omega_{0}^{2}-M\Delta}}-tan^{-1}\dfrac{\sqrt{q^{2}B_{0}^{2}+4M\Delta}}{2\sqrt{M^{2}\omega_{0}^{2}e^{-\Gamma\,t}-M\Delta}}\right]$
$\displaystyle+\dfrac{qB_{0}(n+l)}{2M\Gamma}\,log\,\dfrac{{(4M^{2}\omega_{0}^{2}+q^{2}B_{0}^{2})\left\\{\left(\sqrt{\omega_{0}^{2}(M\sigma-1)+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}e^{\Gamma\,t}}+\dfrac{qB_{0}\sigma}{2}e^{\Gamma\,t/2}\right)^{2}+\omega_{0}^{2}(M\sigma-1)^{2}\right\\}}}{(4M^{2}\omega_{0}^{2}+q^{2}B_{0}^{2}\,e^{\Gamma\,t})\left\\{\left(\sqrt{\omega_{0}^{2}(M\sigma-1)+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}}+\dfrac{qB_{0}\sigma}{2}\right)^{2}+\omega_{0}^{2}(M\sigma-1)^{2}\right\\}}$
$\displaystyle+\dfrac{(n+l)\sqrt{q^{2}B_{0}^{2}+4M\Delta}}{2M\Gamma}\,log\,\dfrac{\left[(4M^{2}\omega_{0}^{2}+q^{2}B_{0}^{2})\left\\{q^{2}B_{0}^{2}\sigma
M^{2}\omega_{0}^{2}+4M^{3}\sigma\Delta\omega_{0}^{2}-2M\omega_{0}^{2}q^{2}B_{0}^{2}-4\omega_{0}^{2}M^{2}\Delta+e^{\Gamma\,t}\left(\dfrac{q^{4}B_{0}^{4}\sigma}{4}+q^{2}B_{0}^{2}\Delta\right)\atop+e^{\Gamma\,t}q^{2}B_{0}^{2}\sigma
M\Delta-2qB_{0}\sqrt{(q^{2}B_{0}^{2}+4M\Delta)\left(\omega_{0}^{2}[M\sigma-1]+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}e^{\Gamma\,t}\right)(M\Delta
e^{\Gamma\,t}-M^{2}\omega_{0}^{2})}\right\\}\right]}{\left[(4M^{2}\omega_{0}^{2}+q^{2}B_{0}^{2}e^{\Gamma\,t})\left\\{q^{2}B_{0}^{2}\sigma
M^{2}\omega_{0}^{2}+4M^{3}\sigma\Delta\omega_{0}^{2}-2M\omega_{0}^{2}q^{2}B_{0}^{2}-4\omega_{0}^{2}M^{2}\Delta+\dfrac{q^{4}B_{0}^{4}\sigma}{4}+q^{2}B_{0}^{2}\Delta\atop+q^{2}B_{0}^{2}\sigma
M\Delta-2qB_{0}\sqrt{(q^{2}B_{0}^{2}+4M\Delta)\left(\omega_{0}^{2}[M\sigma-1]+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}\right)(M\Delta-M^{2}\omega_{0}^{2})}\right\\}\right]}$
$\displaystyle+\dfrac{(n+l)\sqrt{\sigma\Delta}}{\Gamma}\,log\,\dfrac{\left[{\omega_{0}^{2}M^{2}\sigma\Delta-M\Delta\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}\sigma
M\omega_{0}^{2}}{4}+\dfrac{q^{2}B_{0}^{2}\sigma\Delta}{2}e^{\Gamma\,t}\atop+\sqrt{q^{2}B_{0}^{2}\sigma\Delta\left(M\Delta
e^{\Gamma\,t}-M^{2}\omega_{0}^{2}\right)\left(\omega_{0}^{2}[M\sigma-1]+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}e^{\Gamma\,t}\right)}}\right]}{\left[{\omega_{0}^{2}M^{2}\sigma\Delta-M\Delta\omega_{0}^{2}-\dfrac{q^{2}B_{0}^{2}\sigma
M\omega_{0}^{2}}{4}+\dfrac{q^{2}B_{0}^{2}\sigma\Delta}{2}\atop+\sqrt{q^{2}B_{0}^{2}\sigma\Delta\left(M\Delta-M^{2}\omega_{0}^{2}\right)\left(\omega_{0}^{2}[M\sigma-1]+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}\right)}}\right]}+\dfrac{(n+l)qB_{0}}{2M\Gamma}log\,\dfrac{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}e^{-\Gamma\,t}}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}$
$\displaystyle+\dfrac{(n+l)iqB_{0}\sqrt{M\sigma-1}}{2M\Gamma}\times$
$\displaystyle\left[log\dfrac{{M\sigma\Delta-\Delta-\dfrac{q^{2}B_{0}^{2}\sigma}{4}-2M\omega_{0}^{2}(M\sigma-1)e^{-\Gamma\,t}-2i\sqrt{(M\sigma-1)\left(\omega_{0}^{2}[M\sigma-1]e^{-\Gamma\,t}+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}\right)(M\Delta-M^{2}\omega_{0}^{2}e^{-\Gamma\,t})}}}{{M\sigma\Delta-\Delta-\dfrac{q^{2}B_{0}^{2}\sigma}{4}-2M\omega_{0}^{2}(M\sigma-1)-2i\sqrt{(M\sigma-1)\left(\omega_{0}^{2}[M\sigma-1]+\dfrac{q^{2}B_{0}^{2}\sigma}{4M}\right)(M\Delta-M^{2}\omega_{0}^{2})}}}\right]$

We also discussed about the eigenfunctions corresponding to the rationally
decaying EP solution set in section $4.2$. We mentioned that the exact form of
the phase factor to construct the eigenfunction of a damped oscillator having
rationally decaying frequency in the presence of a magnetic field decaying
rationally with respect to time in NC space [Set II, Case I] can be found. The
phase factor is as follows,

$\displaystyle\Theta_{\,n\,,\,l}(t)\,=-\dfrac{2(n+l)}{\Gamma}\left(\dfrac{\sigma}{\mu^{2}}+\dfrac{qB_{0}M\omega_{0}^{2}}{4M^{2}\omega_{0}^{2}+q^{2}B_{0}^{2}}\right)log\,\dfrac{\Gamma\,t+\chi}{\chi}+\dfrac{4(n+l)M^{2}\omega_{0}^{2}}{(q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2})\Gamma}\left[\sqrt{\left(\dfrac{q^{2}B_{0}^{2}\sigma}{M}+4\omega_{0}^{2}\sigma\,M\right)\dfrac{1}{\chi^{2}}-\omega_{0}^{2}}\right.$
$\displaystyle-$
$\displaystyle\left.\sqrt{\left(\dfrac{q^{2}B_{0}^{2}\sigma}{M}+4\omega_{0}^{2}\sigma\,M\right)\dfrac{1}{(\Gamma\,t+\chi)^{2}}-\omega_{0}^{2}}\,+\omega_{0}\left\\{tan^{-1}\dfrac{\sqrt{M}~{}\omega_{0}\chi}{\sqrt{\left(q^{2}B_{0}^{2}\sigma+4\omega_{0}^{2}\sigma\,M^{2}\right)-\omega_{0}^{2}\chi^{2}M}}\right.\right.$
$\displaystyle\left.\left.-tan^{-1}\dfrac{\sqrt{M}~{}\omega_{0}(\Gamma\,t+\chi)}{\sqrt{\left(q^{2}B_{0}^{2}\sigma+4\omega_{0}^{2}\sigma\,M^{2}\right)-\omega_{0}^{2}M(\Gamma\,t+\chi)^{2}}}\right\\}\right]+\dfrac{2(n+l)qB_{0}}{q^{2}B_{0}^{2}+4M^{2}\omega_{0}^{2}}\left[\dfrac{1}{\Gamma}\left\\{\sqrt{\left(\Delta-\dfrac{M\omega_{0}^{2}}{\chi^{2}}\right)\left({4M^{2}\omega_{0}^{2}\sigma+\atop
q^{2}B_{0}^{2}\sigma-M\omega_{0}^{2}\chi^{2}}\right)}\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.\sqrt{\left(M\Delta-\dfrac{M^{2}\omega_{0}^{2}}{(\Gamma\,t+\chi)^{2}}\right)\left(\dfrac{q^{2}B_{0}^{2}\sigma}{M}+4\omega_{0}^{2}\sigma\,M-\omega_{0}^{2}(\Gamma\,t+\chi)^{2}\right)}\right\\}-\dfrac{2iM\omega_{0}^{2}}{\Gamma}\left\\{EllipticE\left(isinh^{-1}\left[\dfrac{i\omega_{0}\sqrt{M}(\Gamma\,t+\chi)}{\sqrt{q^{2}B_{0}^{2}\sigma+4\omega_{0}^{2}\sigma
M^{2}}}\right]\right.\right.\right.$ ,
$\displaystyle\left.\left.\left.\dfrac{q^{2}B_{0}^{2}\sigma\Delta+4\omega_{0}^{2}\sigma
M^{2}\Delta}{M^{2}\omega_{0}^{4}}\right)-EllipticE\left(isinh^{-1}\left[\dfrac{i\omega_{0}\sqrt{M}\chi}{\sqrt{q^{2}B_{0}^{2}\sigma+4\omega_{0}^{2}\sigma
M^{2}}}\right],\dfrac{q^{2}B_{0}^{2}\sigma\Delta+4\omega_{0}^{2}\sigma
M^{2}\Delta}{M^{2}\omega_{0}^{4}}\right)\right\\}\right.$
$\displaystyle-\left.\dfrac{i(q^{2}B_{0}^{2}\sigma\Delta+4\omega_{0}^{2}\sigma
M^{2}\Delta-M^{2}\omega_{0}^{4})}{\Gamma
M\omega_{0}^{2}}\left\\{EllipticF\left(isinh^{-1}\left[\dfrac{i\omega_{0}\sqrt{M}(\Gamma\,t+\chi)}{\sqrt{q^{2}B_{0}^{2}\sigma+4\omega_{0}^{2}\sigma
M^{2}}}\right],\dfrac{q^{2}B_{0}^{2}\sigma\Delta+4\omega_{0}^{2}\sigma
M^{2}\Delta}{M^{2}\omega_{0}^{4}}\right)\right.\right.$ $\displaystyle-$
$\displaystyle\left.\left.EllipticF\left(isinh^{-1}\left[\dfrac{i\omega_{0}\sqrt{M}\chi}{\sqrt{q^{2}B_{0}^{2}\sigma+4\omega_{0}^{2}\sigma
M^{2}}}\right],\dfrac{{q^{2}B_{0}^{2}\sigma\Delta+\atop 4\omega_{0}^{2}\sigma
M^{2}\Delta}}{M^{2}\omega_{0}^{4}}\right)\right\\}\right]+\dfrac{(n+l)}{\Gamma}\left[{\sqrt{\dfrac{\Delta}{M}(\Gamma\,t+\chi)^{2}-\omega_{0}^{2}}-\sqrt{\dfrac{\Delta}{M}\chi^{2}-\omega_{0}^{2}}}\right.$
$\displaystyle+$
$\displaystyle\left.\omega_{0}\,tan^{-1}\sqrt{\dfrac{M\omega_{0}^{2}}{\Delta(\Gamma\,t+\chi)^{2}-M\omega_{0}^{2}}}-\omega_{0}\,tan^{-1}\sqrt{\dfrac{M\omega_{0}^{2}}{\Delta\chi^{2}-M\omega_{0}^{2}}}\right]-\dfrac{(n+l)q^{2}B_{0}^{2}}{M(4M^{2}\omega_{0}^{2}+q^{2}B_{0}^{2})\Gamma}\left[\sqrt{M\Delta(\Gamma\,t+\chi)^{2}-M^{2}\omega_{0}^{2}}\right.$
$\displaystyle-$
$\displaystyle\left.\sqrt{M\Delta\chi^{2}-M^{2}\omega_{0}^{2}}+M\omega_{0}\,\left(tan^{-1}\dfrac{M\omega_{0}}{\sqrt{M\Delta(\Gamma\,t+\chi)^{2}-M^{2}\omega_{0}^{2}}}-\,tan^{-1}\dfrac{M\omega_{0}}{\sqrt{M\Delta\chi^{2}-M^{2}\omega_{0}^{2}}}\right)\right]~{}.$
(89)

Here $EllipticF$ and $EllipticE$ are the incomplete elliptic integrals of the
first and second kinds respectively.

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