Autoregressive Moving Average and Generalized Autoregresive Moving Average in
Covid-19’s Confirmed Cases in Indonesia

Khusnia Nurul Khikmah, A’yunin Sofro*

Mathematics Department, Universitas Negeri Surabaya, East Java, Indonesia

*Corresponding author; e-mail<EMAIL_ADDRESS>
Abstract
Autoregressive moving average and generalized autoregressive moving average
are often used in statistical modeling. This study using this method because
the method uses data from the previous period to model the data for the
current period. In addition, the technique is often used in data prediction.
The familiar data used is count data. Count data is the data that most often
cause data not to spread usually. Therefore, time series modeling, one of
which is through arithmetic series, was developed. This study aims to obtain
the best modeling results from positive confirmed cases of Covid-19 in
Indonesia. They were getting the results from the best modeling for positive
confirmed cases of Covid-19 in Indonesia based on the smallest Aikake’s
information criterion value.

Keywords: Covid-19, Indonesia, autoregressive moving average, generalized
autoregressive moving average, aikake’s information criterion.

##  1\. Introduction

Coronavirus disease 2019 (Covid-19) is an infectious disease caused by the
acute respiratory syndrome coronavirus 2 (SARS-CoV-2) WHO. (n.d.). This
disease was first reported in 2019 in Wuhan, China, and since then, it has
spread globally Hui et al. (2020) including in Indonesia. Indonesia confirmed
for the first time this Covid-19 case on Monday 2 March 2020 News, T. D.
(2020).

General indications of Covid-19 are fever, shortness of breath, cough, muscle
aches, diarrhea, and sore throat Patel A (2020). Covid-19 is usually spread
through the air produced during coughing by one person and can also be spread
from touching surfaces contaminated by the virus Patel A (2020). Covid-19 can
survive on surfaces for up to 72 hours (Health, 2020) dan and the time from
exposure to onset of indication is generally two to fourteen days Patel A
(2020).

Covid-19 is a global pandemic until 30 April 2021. Based on data published by
the World Health Organization (WHO), there are 151517785 confirmed cases from
216 countries WHO202. This global pandemic has made many countries, one of
which is Indonesia, participate in trying to participate in overcoming it is
spread Setiawan, A. R. (2020),Hamid AR (2020).

Covid-19 is an infectious disease that has the potential to cause public
health emergencies. Therefore, the Indonesian Government issued urgency about
the formation of regulations related to the prevention of Covid-19 in a
Government Regulation and Regulation of the Minister of Health Telaumbanua D
(2020).

Based on data published by Indonesian Gugus Tugas Percepatan Penanganan
COVID-19 on April 30, 2021, in Indonesia, as many as 1668368 confirmed
positive for Covid-19 (COVID, 2020). Because this disease spreads very quickly
in a short time, it makes sense to carry out research and analysis on
forecasting the number of COVID-19 cases in the future. Then the generalized
autoregressive moving average ii modeling results hope to be a recommendation
for handling Covid-19 cases based on positive confirmed Covid-19 data in
Indonesia.

Forecasting is processing past data to obtain future data estimates Wulandari
HR (2017). There are many types of data available such as interval, nominal,
and count data. One of the count data is data from a case. Count data is the
data that most often cause data not to spread usually. Therefore, time series
forecasting modeling is developed, one of which is arithmetic series data
Benjamin et al. (2003). Modeling is not only closely related to forecasting
but also closely related to prediction. The prediction itself is something
associated with the outcome of a possibility. Forecasting discusses the
analysis of past data to obtain future options, while prediction is an
analysis of future cases. The models used in this research are autoregressive
moving average and generalized autoregressive moving average. The
autoregressive moving average model consists of two components: Autoregressive
and Moving Average. Autoregressive models the autocorrelation of time series
variables that depend linearly on the values of the previous variables—Moving
Average models the autocorrelation of earlier errors in the time series Hanke
and Wichern (2009). The autoregressive moving average model is a combination
of the autoregressive and moving average models so that it assumes that
previous data influence the current period data and last error values Kasanah
LN. (2016).

Benjamin et al. developed a generalized autoregressive moving average model
for data that follows a non-Gaussian distribution. The generalized
autoregressive moving average model is a development of the expansion of
generalized linear models (GLM) Hanke and Wichern (2009) and the result of
combining autoregressive moving average components with predictor variables to
transform the average parameters of the data distribution using the link
function Benjamin et al. (2003), Hanke and Wichern (2009). In this study,
modeling was carried out using the generalized autoregressive moving average.
The data used is data on the number of positive cases of Covid-19 in Indonesia
from the first confirmed cases of Covid-19 in Indonesia from March 2, 2020, to
April 30, 2021.

##  2\. Literature Review

###  2.1 Autoregressive (AR)

The autoregressive model $AR\left(\textit{p}\right)$ is a model which states
that data in the current period is influenced by data in the previous period
Lilipaly GS (2014). This model is also the most basic model for a stationary
process, or it can also interpreting as a process of regression results by
itself. Mathematically it can be written (Asriawan, 2014), Habibi and Wardhani
(2019):

$\displaystyle
X_{t}=phi_{1}X_{t-1}+\phi_{2}X_{t-2}+...+\phi_{p}X_{t-p}+\phi_{t}.$ (2.1)

The $X_{t}$ is data in $t$ period with $t=1,2,3,…,n$, $X_{t-i}$ is data in
$t-i$ period with $i=1,2,3,…,p$, $\phi_{t}$ is an error in $t$ period, and
$\phi_{i}$ is AR coefficient with $i=1,2,3,…,p$.

###  2.2 Moving Average (MA)

Mathematically, the Moving Average $MA\left(\textit{q}\right)$ model can be
written as follows:

$\displaystyle
X_{t}=e_{t}-\theta_{1}e_{t-1}-\theta_{2}e_{t-2}-...-\theta_{q}e_{t-q}.$ (2.2)

The $X_{t}$ is the predicted variable, $\theta_{1},\theta_{2},...,\theta_{q}$
is parameters of Moving Average, and $e_{t},e_{t-1},e_{t-2},...,e_{t-q}$ is
error value in $t$ period with $t-1,…,t-q$.

###  2.3 Autoregressive Moving Average (ARMA)

The autoregressive moving average is a model known as the autoregressive
integrated moving average model without the ARIMA differentiation process or
$ARIMA\left(p,0,q\right)$. Mathematically $ARMA\left(p,q\right)$ can be
written from 2.1 dan 2.2.

$\displaystyle
X_{t}=phi_{1}X_{t-1}+...+\phi_{p}X_{t-p}+e_{t}-\theta_{1}e_{t-1}-...-\theta_{q}e_{t-q}.$
(2.3)

The $X_{t}$ is data in $t$ period with $t=1,2,3,…,n$,
$\phi_{1},\phi_{2},...,\phi_{p}$ is parameters of Autoregressive,
$\theta_{1},\theta_{2},...,\theta_{q}$ is parameters of Moving Average, and
$e_{t},e_{t-1},e_{t-2},...,e_{t-q}$ is error value in $t$ period with
$t-1,…,t-q$. The autoregressive moving average model in time series data
applies the Box-Jenkins procedure. The first is model identification. Model
identification in autoregressive moving average can see from the
autocorrelation function and partial autocorrelation function plots used to
determine the p and q orders in the $ARMA\left(p,q\right)$ model. The second
is parameter estimation. Parameter estimation of the model can use the maximum
likelihood estimation (MLE) method. The third is forecasting. Forecasting is
the final stage of the analysis using time series data. Then, diagnostic
check. And the last is the accuracy model.

###  2.4 Autocorrelation Function (ACF) and Partial Autocorrelation Function
(PACF)

The autocorrelation function is a linear relationship indicator on
observations. For example, the observations $Z_{t}$ and $Z_{t+k}$. While the
partial autocorrelation function is an indicator of the magnitude of the
relationship between the values of the same variable with other time delays
that have an effect that is considered constant. Autocorrelation function and
partial autocorrelation function have two patterns, namely cut off and dies
down. When the autocorrelation function and partial autocorrelation function
lines are significant in the first lag, but in the subsequent lag, there are
not many autocorrelation functions and partial autocorrelation function lines,
the pattern is included in the cut off pattern. Meanwhile, if the
autocorrelation function and partial autocorrelation function lines decrease
gradually or are not cut, the pattern is included in the dies down pattern.
Suppose the autocorrelation function is $\hat{\rho}{{}_{k}}$, then:

$\displaystyle\hat{\rho}{{}_{k}}=\frac{\sum_{t=1}^{n-k}\left(Z_{t}-Z\right)\left(Z_{t+k}-Z\right)}{\sum_{t=1}^{n-k}\left(Z_{t}-Z\right)^{2}}.$
(2.4)

And partial autocorrelation function is $\hat{\phi}{{}_{kk}}$ with
$j=1,2,3,…,k-1$ and
$\hat{\phi}{{}_{kj}}=\hat{\phi}{{}_{k-1,j}}-\hat{\phi}{{}_{kk}}\hat{\phi}{{}_{k-1,k-j}}$,
then:

$\displaystyle\hat{\phi}{{}_{kk}}=\frac{r_{k}-\sum_{j=1}^{k-1}\hat{\phi}_{{k-1,j}}r_{k-j}}{1-\sum_{j=1}^{k-1}\hat{\phi}_{{k-1,j}}r_{j}}.$
(2.5)

###  2.5 Generalized Autoregressive Moving Average (GARMA)

The generalized autoregressive moving average model was first introduced by
Benjamin Benjamin et al. (2003):

$\displaystyle g\left(\mu\right)=Z_{t-1}^{T}\beta=X_{t-1}^{T}\beta+\tau_{t}.$
(2.6)

With

$\displaystyle\tau_{t}=\sum_{j=1}^{p}\phi_{j}A\left(y_{t-j},x_{t-j},\beta\right)+\sum_{j=1}^{q}\theta_{j}M\left(y_{t-j},\mu_{t-j}\right).$
(2.7)

The $\tau_{t}$ is components of AR and MA, $A$ is a function that represents
the Autoregressive form, $M$ is a function that represents the moving average
form, $\phi^{T}$ is Autoregressive parameter with
$\phi^{T}=\left(\phi_{1},\phi_{2},...,\phi_{p}\right)$, and $\theta^{T}$ is
Moving Average parameter
$\theta^{T}=\left(\theta_{1},\theta_{2},...,\theta_{q}\right)$.

###  2.6 Maximum Likelihood Estimation (MLE)

Maximum likelihood estimation is one method that functions as an estimation
method that maximizes the likelihood function. Mathematically it can be
written Benjamin et al. (2003), (Asriawan, 2014):

$\displaystyle
L\left(\theta\right)=\prod_{t=1}^{n}f\left(x_{i}\mid\theta\right).$ (2.8)

Calculating the maximum likelihood estimation can be made easier by using the
log-likelihood function, mathematically can be written:

$\displaystyle
l\left(\theta\right)=lnL\left(\theta\right)=\sum_{i=1}^{n}lnf\left(x_{i}\mid\theta\right).$
(2.9)

###  2.7 Validation

####  2.7.1 Mean Absolute Error (MAE) and Root Mean Square Error (RMSE)

Mean absolute error and root mean square error are validations that are often
used to compare and evaluate the accuracy of the results of a prediction from
an equation with actual data. Suppose the $(n)$ data with $(e_{i})$ is the
error of the model, which is the result of subtraction from the predicted
value $(x_{i})$ with the actual value $(y_{i})$ and $i=1,2,3,…,n$, then the
mean absolute error value is:

$\displaystyle
MAE=\frac{1}{n}\sum_{i=1}^{n}\left|x_{i}-y_{i}\right|=\frac{1}{n}\sum_{i=1}^{n}\left|e_{i}\right|.$
(2.10)

And the root mean square error value is:

$\displaystyle RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left|e_{i}\right|}.$
(2.11)

####  2.7.2 Aikaike’s Information Criteria (AIC)

Aikake’s information criterion was first introduced in 1973 by Aikake. Value
Aikake’s information criterion serves as a conduit of relative size
suitability pointer statistical models. In short, the value of Aikake’s
information criterion is an indicator of the size of the fit of the model with
the data being modeled Ray S (2015). In addition, the value of Aikake’s
information criterion also minimizes the loss of information in model
selection and provides information to determine the best model. Where the best
model obtained is obtained based on the smallest value of Aikake’s information
criterion Ray S (2015). Value Aikake’s information criterion can be obtained
by are the parameters of the model and is the maximum value of the likelihood
function that is used to estimate the model and defined asAcquah HD
(2010),Anderson et al. (1998), Ray S (2015):

$\displaystyle AIC=2k-2ln\left(L\right).$ (2.12)

##  3\. Result and Discussion

This study took secondary data from the official website of the task force for
the acceleration of handling Covid-19 in Indonesia. The data to be processed
is data on positive cases of Covid-19 per day in Indonesia from March 2, 2020,
to April 30, 2021, with 425 data. This data has a plot in the time series data
presented in Figure 1.

Figure 1: Time Series Plot of Data Positive Cases of COVID-19 Per Day in
Indonesia from March 2, 2020, to April 30, 2021

The first stage in formulating an autoregressive moving average model is
identifying data according to the first stage in data analysis. The
descriptive statistics from the data are attaching in Table 1.

Table 1: Descriptive Statistics of Data Positive Cases of COVID-19 Per Day in Indonesia from March 2, 2020, to April 30, 2021 No | Statistic Descriptive of Data | Value
---|---|---
1 | Min. | 0
2 |  $1^{st}$ Quartil | 1041
3 | Median | 3622
4 | Mean | 3926
5 |  $3^{rd}$ Quartil | 5803
6 | Max. | 14518
7 | Sd. | 3344.91
8 | Var. | 11188420

The second stage of data analysis is dividing the data into train data and
test data. Train data will be processed using the ARMA model with a ratio of
90% of the data used as train data, and 10% of the data will be used as test
data. Then the divided data, namely the train data, is tested for stationary.
In this study, the stationary using Augmented Dickey-Fuller test. The
augmented dickey-fuller test has a hypothesis if the $p-value>\alpha$ shows
that the data is not stationary or rejects the initial hypothesis, so there is
a need to do a differences process. If $p-value<\alpha$ shows that the data is
stationary, there is no need to do a differences process. In this study,
$\alpha$ used is $\alpha=0.05$ or 5%. The augmented dickey-fuller test from
the data, which has a $p-value$ or $p-value=0.829>\alpha=0.05$, means the data
is not yet stationary or rejects the initial hypothesis, so there is a need to
do a differences process. The augmented dickey-fuller test from the data,
after differences process, has a $p-value$ or $p-value=0.01<\alpha=0.05$,
which means the data is stationary or accepts the initial hypothesis, so there
is a need to do a differences process.

Time series plot of stationary data is attaching on Figure 2 and The
stationary time series data can be seen from the qqnorm plot, which is linked
in Figure 3.

Figure 2: Time Series Plot of Stationary Data Positive Cases of COVID-19 Per
Day in Indonesia Figure 3: Q-Q Plot of Stationary Data Positive Cases of
COVID-19 Per Day in Indonesia

The third step in data analysis is forecasting the ARMA model. The first step
in forecasting the ARMA model is identifying the model. The aim of identifying
the model is to determine the order in the ARMA model. The order is p and q,
obtained from the autocorrelation function and partial autocorrelation
function plots. The autocorrelation function and partial autocorrelation
function plots are attaching in Figure 4 and 5.

Figure 4: Autocorrelation function Plot of Stationary Data Positive Cases of
COVID-19 Per Day in Indonesia Figure 5: Partial autocorrelation function Plot
of Stationary Data Positive Cases of COVID-19 Per Day in Indonesia

Based on the autocorrelation function and partial autocorrelation function
from Figure 4 and 5 a model formed from data on positive cases of Covid-19 per
day in Indonesia from March 2, 2020, to April 30, 2021, is an ARMA(1,1) and
MA(5) and the model also be obtained by ARIMA(1,0,1) and ARIMA(0,0,5). From
the ARMA(1,1) and MA(5) models, the forecasting results were obtained having
an RMSE value of $850,654$, MAE of $497,2133$, and an AIC value of $6249$ for
the ARMA(1,1) model and the MA(5) model having RMSE value is $1193,662$, MAE
is $820,0247$, and AIC is $6512.66$ forecasting results with the ARMA(1,1),
and MA(5) has a plot as in Figure 6 and Figure 7.

Figure 6: Forecasting Plot of ARMA(1,1) Model Figure 7: Forecasting Plot of
MA(5) Model

The modeling results are ARMA(1,1) and MA(5) copy can be used for forecasting
can be used for prediction. Figures 8 and 9 show the prediction results of the
ARMA(1,1) and MA(5) models against the original data. This prediction result
has a red plot and a black color for the original data.

Figure 8: Prediction Plot of ARMA(1,1) Model Figure 9: Prediction Plot of
MA(5) Model

Based on the error values, namely the Root Mean Square Error (RMSE), Mean
Absolute Error (MAE), and AIC values and the results of forecasting and
predictions, it is obtained that the best model for data on positive cases of
COVID-19 in Indonesia is ARMA(1,1). The residual ARMA(1,1) model has checked
by diagnostic check uses the Ljung-Box method with the hypothesis that if
$p-value>\alpha$ indicates that the residual is independently distributed and
if the $p-value<\alpha$ means that the residual of the model not independently
distributed (serial correlation). The result of the Ljung Box of ARMA(1,1)
model has $0.04591$, $\alpha$ used is equal to $\alpha=0.05$ or 5%, which
means that the null hypothesis is rejected. We can conclude that the residuals
are independently distributed, or it shows that with ARMA(1,1) model, the
assumption of making a mistake is 5%. The residual of the ARMA(1,1 ) model is
independent at the 95% level. ARMA(1,1) model is a good fit model and has a
residual result:

Figure 10: The Residual Plot of ARMA(1,1) Model

The best previous modeling result based on the smallest error value obtained
is ARMA(1,1). This ARMA(1,1) model will then be used for calculations on GARMA
with the GARMA(1,1) model calculated. The results of GARMA(1,1) modeling are
obtained:

Table 2: The results of modeling with the GARMA(1,1) model No | estimate | Std Error
---|---|---
1 | intercept | -4.64217 | 0.031575
2 | lag | 1.719924 | 0.001682

The modeling results, namely through the ARMA model in the form of forecasting
and predictions, then the results of the GARMA ii modeling have the hope of
being a recommendation for handling Covid-19 cases based on positive confirmed
Covid-19 data in Indonesia. This research focuses on data confirmed to be
positive for Covid-19 in Indonesia because from the start of the pandemic
until now, positive cases of Covid-19 in Indonesia are still the main focus,
and based on data, positive cases of Covid-19 in Indonesia are still
increasing. The modeling in this study includes time series count data that
uses two models, namely ARMA and GARMA. Research on time series count data not
only uses the ARMA and GARMA models, but some of its developments are also
available through ARMA-GARCH, Bayesian GARMA, and others.

##  4\. Conclusions

Research that focuses on data on positive cases of Covid-19 in Indonesia from
March 2, 2020, to April 30, 2021, which are time-series data, was forecasted
and predicted to obtain two models. The models used are ARMA and GARMA with
the hope of being a recommendation for handling Covid-19 cases based on
positive confirmed Covid-19 data in Indonesia. The modeling results obtained
are the ARMA(1.1) model with an RMSE value of 850,654, MAE of $497,2133$, AIC
value of $6249$ and MA(5) with a value of RMSE of $1193,662$, MAE of
$820,0247$, and AIC of $6512.66$. Of the two models, the best model obtained
is ARMA(1,1), with a smaller error value than the MA(5) model. The best model
obtained by ARMA(1,1) is continued to modeling with GARMA(1,1). The GARMA(1.1)
model produces parameter estimates of $-4.64217$ and $1.719924$ with errors of
$0.031575$ and $0.001682$, respectively. The modeling results in this study
can also be continued through the ARMA-GARCH model and the Bayesian GARMA
model.

## Acknowledgements

We would like to thank the referees for his comments and suggestions on the
manuscript.

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# A novel isospectral deformation chain in supersymmetric quantum mechanics

Bjørn Jensen  Faculty of Micro and Nano Systems Technology, Vestfold
University College, N-3103 Tønsberg Norway<EMAIL_ADDRESS>
###### Abstract

We present a novel isospectral deformation chain in supersymmetric quantum
mechanics. The chain is applied to the Coulomb potential.

###### pacs:

03.65.Ge, 02.40.-k, 42.50.-p

##  I Introduction

In Jensen  we introduced within the framework of supersymmetric quantum
mechanics an isospectral deformation on the form

$W(x)\rightarrow\hat{W}_{0}(x)=F_{0}(x)W(x)\,,$ (1)

where $W(x)$ is some known superpotential and $F_{0}(x)$ some function to be
determined by the isospectrality condition

$\hat{V}_{+}(x)\equiv\hat{W}_{0}^{2}(x)+\hat{W}_{0}^{\prime}(x)=W^{2}(x)+W^{\prime}(x)\equiv
V_{+}(x)\,.$ (2)

It was shown that Eq.(1) includes the only previously explored deformation of
this kind, which has the form (Asim , e.g.)

$W(x)\rightarrow\hat{W}_{0}(x)=W(x)+f(x)\,.$ (3)

$f(x)$ is some function which is determined by Eq.(2). In this paper we expand
the deformation Eq.(1) in various directions and study the implications drawn
from the isospectrality condition. We show in particular that the deformation
Eq.(1) is the root of an infinitely long and recursively generated chain of
deformations.

In the next section we briefly review some findings in Jensen . We then define
and derive an explicit recursive scheme for generating novel isospectral
deformations. We then discuss possible generalizations of this scheme. We next
apply the recursive scheme to the Coulomb potential. We conclude and discuss
some of our findings in the last section.

##  II Base deformations

The deformation Eq.(1) implies the following differential equation for
$F_{0}(x)$ Jensen 111We will often rewrite fractions on the form
$W^{\prime}(x)/W(x)$ as the logarithmic derivative of $W(x)$ as a formal tool.
Caution must of course be exercised when using the corresponding expressions
in actual computations.

$\displaystyle\frac{d}{dx}F_{0}(x)$ $\displaystyle+$
$\displaystyle(\frac{d}{dx}\ln
W(x))F_{0}(x)+W(x)F_{0}^{2}(x)=W(x)+\frac{d}{dx}\ln W(x)\,.$ (4)

This is the generalized Riccati equation Matte  . If one particular solution
$F_{00}(x)$ of Eq.(4) is known another solution is given by Boyce

$F_{0}(x)=F_{00}(x)+\frac{1}{X_{0}(x)}\,,$ (5)

where $X_{0}(x)$ solves the equation

$\frac{d}{dx}X_{0}(x)-(\frac{d}{dx}\ln W(x)+2F_{00}(x)W(x))X_{0}(x)=W(x)\,.$
(6)

Eq.(6) can be solved by elementary means. The resulting superpotential
$\hat{W}_{0}(x)$ is given by Jensen

$\displaystyle\hat{W}_{0}(x)$ $\displaystyle=$
$\displaystyle(F_{00}(x)+\frac{1}{X_{0}(x)})W(x)\equiv\hat{W}_{00}(x)+\frac{1}{X_{0}(x)}W(x)=$
(7) $\displaystyle=$ $\displaystyle
F_{00}(x)W(x)+\frac{e^{-2\int^{x}F_{00}(t)W(t)dt}}{C_{01}+\int^{x}e^{-2\int^{u}F_{00}(t)W(t)dt}du}\,.$

$C_{01}$ is an integration constant, which we will assume to be real. We have
explicitly introduced upper integration limits in Eq.(7) in order to avoid
sign ambiguities. This explains the difference in the sign in the denominator
in Eq.(7) compared with (2.5) in Jensen  where the reverse order of
integration in one of the integrals was implicitly assumed. We do not specify
the lower integration limits in Eq.(7). These are not important, of course,
since the values of the integrals there can essentially be absorbed into
$C_{01}$. When $F_{00}(x)=1$ we identically rederive Eq.(3) and the
corresponding expression discussed in Asim . We can by simple inspection see
that the particular solution $F_{0}(x)=1$, the identity deformation, solves
Eq.(4). This deformation corresponds to the limit $C_{01}\rightarrow\infty$ in
Eq.(7) with $F_{00}(x)=1$. In the limit $C_{01}\rightarrow\infty$ we generally
get $\hat{W}_{0}(x)=\hat{W}_{00}(x)$. This deformation will play a pivotal
role in this work; it will represent the base of a recursive scheme for
generating novel isospectral deformations. We will therefore refer to a
particular $\hat{W}_{00}(x)$ as a base deformation in the following.

In order to expand the space of concrete isospectral deformations further we
transform Eq.(4) into an ordinary second order differential equation by the
substitution

$F_{0}(x)=\frac{1}{W(x)}\frac{d}{dx}\ln U_{0}(x)\,.$ (8)

This substitution gives rise to the following linear homogeneous equation

$-\frac{d^{2}}{dx^{2}}U_{0}(x)+V_{+}(x)U_{0}(x)=0\,.$ (9)

The special solution $F_{0}(x)=1$ is generated by the solution

$U_{0}(x)\sim e^{\int^{x}W(t)dt}\,.$ (10)

The particular solutions for $F_{0}(x)$ stemming from these equations can be
fed into Eq.(7) (as $F_{00}(x)$) and thus expand the space of available
concrete deformations. The physical potential $\hat{V}_{-}(x)$ generated by
$\hat{W}_{0}(x)$ can in general thus be written Jensen 222Note that the
corresponding expression in Jensen  ((2.14)) is misprinted.

$\displaystyle\hat{V}_{-}(x)$ $\displaystyle\equiv$
$\displaystyle\hat{W}_{0}^{2}(x)-\hat{W}_{0}^{\prime}(x)=\hat{W}_{00}^{2}(x)-\hat{W}_{00}^{\prime}(x)+$
(11) $\displaystyle+$
$\displaystyle\frac{4\hat{W}_{00}(x)e^{-2\int^{x}\hat{W}_{00}(t)dt}}{C_{01}+\int^{x}e^{-2\int^{u}\hat{W}_{00}(t)dt}du}+2\left[\frac{e^{-2\int^{x}\hat{W}_{00}(t)dt}}{C_{01}+\int^{x}e^{-2\int^{u}\hat{W}_{00}(t)dt}du}\right]^{2}$

with

$\hat{W}_{00}(x)=\frac{d}{dx}\ln U_{0}(x)\,.$ (12)

##  III Recursive linear deformations

Although the Riccati equation can be transformed into an ordinary second order
differential equation the non-linearity of the equation allows for a solution
space which is larger than the one associated with linear differential
equations of second order, as became evident in the previous section. It is
therefore natural to ask whether the non-linearity of the Riccati equation
implies even more isospectral deformations than the ones we already have
deduced Jensen . We will explore this question in this and the next section.

###  III.1 The sum

Let us entertain the following idea. Assume that we have derived a particular
base deformation $\hat{W}_{00}(x)$ from an explicitly given superpotential
$W(x)$. Then assume that we add another term $F_{1}(x)W(x)$ to that
deformation such that we in principle get a novel deformation on the form
$\hat{W}(x)=F_{10}(x)W(x)+\hat{W}_{00}(x)$. After determining $F_{10}(x)$ from
the isospectrality condition Eq.(2) add yet another term of this kind to the
deformation. Let us assume that this process can be repeated indefinitely.
Will terms added in this manner give rise to novel deformations? We will in
the following show that they do. This represents a recursive deformation
scheme.

Following the basic idea, after $m$ iterations we thus have the general
recursive linear (in $W(x)$) deformation

$\hat{W}_{m0}(x)=(\sum_{i=0}^{m}\lambda_{i}F_{i0}(x))W(x)=\lambda_{m}F_{m0}(x)W(x)+\hat{W}_{(m-1)0}(x)\,\,,\,\,\lambda_{0}\equiv
1\,.$ (13)

The $\lambda_{i}$’s are assumed to be independent real constants. Starting
with a known superpotential $m$ consecutive applications of the isospectrality
condition yields the following set of equations

$\left\\{\begin{array}[]{l}F_{00}^{\prime}(x)+[\ln
W(x)]^{\prime}F_{00}(x)+W(x)F_{00}^{2}(x)=W(x)+(\ln W(x))^{\prime}\,,\\\
F_{10}^{\prime}(x)+[(\ln
W(x))^{\prime}+2F_{00}(x)W(x)]F_{10}(x)+\lambda_{1}W(x)F_{10}^{2}(x)=0\,,\\\
F_{20}^{\prime}(x)+[(\ln
W(x))^{\prime}+2(F_{00}(x)+\lambda_{1}F_{10}(x))W(x)]F_{20}(x)+\lambda_{2}W(x)F_{20}^{2}(x)=0\,,\\\
\vdots\\\ \vdots\\\ F_{m0}^{\prime}(x)+[(\ln
W(x))^{\prime}+2\hat{W}_{(m-1)0}(x))]F_{m0}(x)+\lambda_{m}W(x)F_{m0}^{2}(x)=0\,.\end{array}\right.$
(14)

The first equation in Eq.(14) coincides of course per definition with Eq.(4).
Note that $F_{j0}(x)=1$ only solves the first equation in Eq.(14). Let us
consider an arbitrary iteration level $n\,(\neq 0)$ and make the following
substitution in Eq.(14)

$F_{n0}(x)=\frac{1}{W(x)}(\ln U_{n}(x))^{\prime}\,.$ (15)

The equation for $F_{n}(x)$ can then be written

$U^{\prime\prime}_{n}(x)+(\lambda_{n}-1)\frac{[U^{\prime}_{n}(x)]^{2}}{U_{n}(x)}+2\hat{W}_{(n-1)0}(x)U^{\prime}_{n}(x)=0\,.$
(16)

This equation corresponds to Eq.(9) in the case when $n=0$. It reduces in
general to an ordinary linear differential equation only when
$\lambda_{n}=1\,,\,\forall n\neq 0$. We will focus on this special case in
this paper.

The general solution of Eq.(16) for arbitrary $n\neq 0$, and with
$\lambda_{n}$ set to unity, can be found by elementary means, and we deduce
that

$\displaystyle F_{n0}(x)W(x)$ $\displaystyle=$
$\displaystyle\frac{C_{n2}e^{-2\int^{x}\hat{W}_{(n-1)0}(t)dt}}{C_{n1}+C_{n2}\int^{x}e^{-2\int^{u}\hat{W}_{(n-1)0}(t)dt}du}=$
(17) $\displaystyle=$
$\displaystyle\frac{d}{dx}\ln(C_{n1}+C_{n2}\int^{x}e^{-2\int^{u}\hat{W}_{(n-1)0}(t)dt}du)\,.$

$C_{n1}$ and $C_{n2}$ are integration constants, which we assume to be real.
We can reduce the number of integration constants to one at each iteration
level, but we will stick to the habit of explicitly writing down the actual
number of constants in order to make it easier to compare the various formulas
we deduce, which stem from both second and first order differential equations.
We also note that the structure of $F_{n0}(x)$ implies that previous
deformations are not regenerated in general. Of course, this does not exclude
this possibility to arise, as we will see in Section 5. Hence, $m$ in Eq.(13)
has in principle no natural upper bound. From Eq.(13) and Eq.(17) we get the
following expression for the superpotential at iteration level $m$

$\displaystyle\hat{W}_{m0}(x)$ $\displaystyle=$
$\displaystyle\hat{W}_{00}(x)+\sum_{j=1}^{m}\frac{d}{dx}\ln(C_{j1}+C_{j2}\int^{x}e^{-2\int^{u}\hat{W}_{(j-1)0}(t)dt}du)=$
(18) $\displaystyle=$
$\displaystyle\hat{W}_{00}(x)+\frac{d}{dx}\ln\prod_{j=1}^{m}(C_{j1}+C_{j2}\int^{x}e^{-2\int^{u}\hat{W}_{(j-1)0}(t)dt}du)\equiv$
$\displaystyle\equiv$ $\displaystyle\hat{W}_{00}(x)+\frac{d}{dx}\ln
P_{m}(x)\,.$

From Eq.(18) we deduce that

$e^{-2\int^{x}\hat{W}_{(j-1)0}(t)dt}=P_{j-1}^{-2}(x)e^{-2\int^{x}\hat{W}_{00}(t)dt}\,\,;\,\,P_{0}^{2}(x)\equiv
1\,,\,j\neq 0\,,$ (19)

such that

$\left\\{\begin{array}[]{l}P_{1}(x)=C_{11}+C_{12}\int^{x}e^{-2\int^{u}\hat{W}_{00}(t)dt}du\,,\\\
P_{2}(x)=P_{1}(x)(C_{21}+C_{22}\int^{x}P_{1}^{-2}(u)e^{-2\int^{u}\hat{W}_{00}(t)dt}du)\,,\\\
\vdots\\\ \vdots\\\
P_{m}(x)=P_{m-1}(x)(C_{m1}+C_{m2}\int^{x}P_{m-1}^{-2}(u)e^{-2\int^{u}\hat{W}_{00}(t)dt}du)\,.\end{array}\right.$
(20)

Hence,

$\displaystyle
P_{n}(x)=\prod_{j=1}^{n}(C_{j1}+C_{j2}\int^{x}P_{j-1}^{-2}(u)e^{-2\int^{u}\hat{W}_{00}(t)dt}du)\,.$
(21)

This last form of the $P_{n}(x)$ functions neatly exhibits how the base
deformation $\hat{W}_{00}(x)$ generates the higher order deformations. Some of
the details we have deduced so far in this subsection are presented in Figure
1.

$\left\\{\begin{CD}W@>{\times
F_{0}=F_{00}}>{}>\hat{W}_{00}@>{+F_{10}W}>{}>\hat{W}_{10}=\hat{W}_{00}+(\ln
P_{1})^{\prime}@>{+F_{20}W}>{}>\cdots @>{+F_{m0}W}>{}>\hat{W}_{m0}\\\ \\\
W@>{\times F_{0}=1}>{}>\hat{W}_{00}=W@>{+F_{10}W}>{}>\hat{W}_{10}=W+(\ln
P_{1})^{\prime}@>{+F_{20}W}>{}>\cdots
@>{+F_{m0}W}>{}>\hat{W}_{m0}\end{CD}\right.$
Figure 1: The upper line depicts the solvable deformation chain Eq.(13) to
iteration level $m$. There is no upper bound on $m$. The $F_{j0}(x)$ functions
are given in Eq.(17). The $P_{j}(x)$ functions are given in Eq.(18) and
Eq.(20). They are functions of a base deformation $\hat{W}_{00}(x)$. A base
deformation $\hat{W}_{00}(x)$ is generated by the zero-energy Schrödinger
equation interacting with the partner potential $V_{+}(x)$, Eq.(9). The second
line depicts the important special case when $F_{0}(x)=1$. This particular
solution can be derived as a special case of Eq.(7) with $X_{0}^{-1}=0$, which
can be achieved by $C_{01}\rightarrow\infty$, and $F_{00}(x)$ determined by
Eq.(8) and the solution Eq.(10). Note that $\hat{W}_{10}(x)$ then coincides
with Eq.(7) when $F_{00}(x)=1$ and $C_{01}$ is finite in Eq.(7); Eq.(7) is
thus in this particular case regenerated by the scheme at the next recursion
level, i.e.

Make the following substitution at each iteration level in Eq.(13)

$F_{n0}(x)\rightarrow F_{n0}(x)+\frac{1}{X_{n}(x)}\,.$ (22)

This implies (with the $\lambda_{m}$’s reinstated in Eq.(13)) a generalized
form $\hat{W}_{m}(x)$ of the superpotential $\hat{W}_{m0}(x)$

$\hat{W}_{m0}(x)=\sum_{i=0}^{m}\lambda_{i}F_{i0}W(x)\Rightarrow\hat{W}_{m}(x)=\hat{W}_{m0}(x)+\sum_{i=0}^{m}\frac{\lambda_{i}}{X_{i}(x)}W(x)\,.$
(23)

From Eq.(14) we find that $X_{n}(x)$ satisfies the equation

$\frac{d}{dx}X_{n}(x)-(\frac{d}{dx}\ln
W(x)+2\hat{W}_{n0}(x))X_{n}(x)=\lambda_{m}W(x)\,.$ (24)

This equation is a generalization of Eq.(6). The $n$’th deformation term
Eq.(17) thus changes into

$\displaystyle F_{n0}(x)W(x)\rightarrow
F_{n0}(x)W(x)+\frac{d}{dx}\ln(C_{n3}+\lambda_{n}\int^{x}e^{-2\int^{u}\hat{W}_{n0}(t)dt}du)\,.$
(25)

$C_{n3}$ are integrations constants, which we assume to be real. Eq.(25)
implies that the more general expression for the superpotential in Eq.(23) can
be written as

$\hat{W}_{m}(x)=\hat{W}_{m0}(x)+\frac{d}{dx}\ln Q_{m}(x)\,,$ (26)

where

$Q_{m}(x)\equiv\prod_{i=0}^{m}(C_{i3}+\lambda_{i}\int^{x}e^{-2\int^{u}\hat{W}_{i0}(t)dt}du)\,.$
(27)

$m=0$ in Eq.(26) ($\lambda_{0}\equiv 1$) reproduces Eq.(7). In the special
case when $\lambda_{m}=1\,,\forall m$ in Eq.(13) we get

$\hat{W}_{m}(x)=\hat{W}_{00}(x)+\frac{d}{dx}\ln P_{m}(x)+\frac{d}{dx}\ln
Q_{m}(x)\,.$ (28)

When we compare the expressions for $(P_{m}(x))^{\prime}$ and
$(Q_{m}(x))^{\prime}$ we find that they differ by just the last term in
$(Q_{m}(x))^{\prime}$.

###  III.2 The product

What happens if we in Eq.(13) assume a product structure instead of a sum
structure ? Let us assume that we have determined a base deformation. Let this
be the seed superpotential for the deformation

$\hat{W}_{00}(x)\rightarrow\hat{W}_{10}(x)=F_{10}(x)\hat{W}_{00}(x)=F_{10}(x)F_{00}(x)W(x)\,,$
(29)

where $F_{10}(x)$ is some function to be determined by the isospectrality
condition. This product scheme can of course in principle be repeated an
arbitrary number $m$ times

$\hat{W}_{m0}(x)=(\prod_{i=0}^{m}F_{i0}(x))W(x)=F_{m0}(x)\hat{W}_{(m-1)0}\,.$
(30)

This structure gives rise to the following set of equations

$\left\\{\begin{array}[]{l}F_{00}^{\prime}(x)+(\ln
W(x))^{\prime}F_{00}(x)+W(x)F_{00}^{2}(x)=W(x)+(\ln W(x))^{\prime}\,,\\\
F_{10}^{\prime}(x)+(\ln\hat{W}_{00}(x))^{\prime}F_{10}(x)+\hat{W}_{00}(x)F_{10}^{2}(x)=\frac{1}{F_{00}(x)}(W+(\ln
W(x))^{\prime})\,,\\\ \vdots\\\ \vdots\\\
F_{m0}^{\prime}(x)+(\ln\hat{W}_{(m-1)0}(x))^{\prime}F_{m0}(x)+\hat{W}_{(m-1)0}(x)F_{m0}^{2}(x)=\frac{W(x)}{\hat{W}_{(m-1)0}(x)}(W(x)+(\ln
W(x))^{\prime})\,.\end{array}\right.$ (31)

Clearly, each iteration level depends on all the previous ones, and at each
level we are dealing with a non-homogenous non-linear differential equation.
Interestingly, by making the following substitution at an arbitrary iteration
level $n\neq 0$

$F_{n0}(x)=\frac{1}{\hat{W}_{(n-1)0}(x)}(\ln U_{n}(x))^{\prime}\,,$ (32)

where $U_{n}(x)$ is some function, the equations Eq.(31) all reduce to Eq.(9).
Hence, attempting to generate novel deformations recursively via a product
structure, of the kind above, fails. This conclusion was also reached in
Jensen , but at the level of the second order linear differential equation
Eq.(9).

##  IV Recursive non-linear deformations

We have so far only considered linear deformations. In this section we will
briefly consider two non-linear deformation schemes. Let us first consider a
polynomial kind of deformation. That is, given a superpotential
$\hat{W}_{(i-1)0}(x)$ which we will assume is derived, in some way or another,
from some seed superpotential $W(x)$. Consider then the polynomial deformation

$\hat{W}_{i0}(x)=F_{i0}(x)W^{k}(x)+\hat{W}_{(i-1)0}(x)\,;\,k\in\\{1,2,3,\ldots\\}\,.$
(33)

The isospectrality condition implies

$F_{i0}^{\prime}(x)+[k(\ln
W(x))^{\prime}+2\hat{W}_{(i-1)0}(x)]F_{i0}(x)+W^{k}(x)F_{i0}^{2}(x)=0\,.$ (34)

This is a Riccati type equation of the kind we have met earlier in this paper.
Apparently, different $k$-values give rise to very different equations to
solve. However, and rather intriguingly, all the possible $k$-values implies
the same deformation. This is seen by making the following substitution

$F_{i0}(x)=\frac{1}{W^{k}(x)}\frac{U^{\prime}_{k}(x)}{U_{k}(x)}\,,$ (35)

where $U_{k}(x)$ is some function. This expression inserted into Eq.(34) gives

$U^{\prime\prime}_{k}(x)+2\hat{W}_{i-1}(x)U^{\prime}_{k}(x)=0\,.$ (36)

Hence, $\hat{W}_{i0}(x)$ is independent of $k$ and we are essentially left
with a linear deformation. Clearly, the range of values of $k$ can be expanded
to the real numbers.

Another canonical generalization of our work is to consider deformations on
the form

$\hat{W}(x)=H_{0}(x){\cal F}(W)\,,$ (37)

where ${\cal F}$ is any functional of the seed superpotential $W(x)$. The
isospectrality condition then implies

$H^{\prime}_{0}(x)+(\ln{\cal F}(W))^{\prime}H_{0}(x)+{\cal
F}(W)H_{0}^{2}(x)={\cal F}(W)^{-1}(W^{2}(x)+W^{\prime}(x))\,.$ (38)

Note that $H_{0}(x)=1$ does not solve this equation unless ${\cal F}(W)=W$,
since Eq.(38) with $H_{0}(x)=1$ implies ${\cal F}^{\prime}(x)+{\cal
F}^{2}(x)=V_{+}(x)$. Since $V_{+}(x)$ is uniquely given in Eq.(2) any other
choice of functional will fail to satisfy the isospectrality condition. Hence
the conclusion. The particular solution $H_{0}(x)=1$ is not forced upon us. We
can in principle do without it. It is easily verified that Eq.(38) can be cast
into the form Eq.(9) by the substitution

$H_{0}(x)=\frac{1}{{\cal F}(W)}(\ln U(x))^{\prime}\,.$ (39)

We can also look for an expanded solution by writing

$H_{0}(x)=H_{00}(x)+\frac{1}{Z_{0}(x)}\,,$ (40)

where $H_{00}(x)$ is a particular solution of Eq.(38). We then get the
equation

$\frac{d}{dx}Z_{0}(x)-(\frac{d}{dx}\ln{\cal F}(W)+2H_{00}(x){\cal
F}(W))Z_{0}(x)={\cal F}(W)\,,$ (41)

which is a generalized form of Eq.(6). The reciprocal solution has the general
form

$\frac{1}{Z_{0}(x)}=\frac{e^{-2\int^{x}H_{00}(t){\cal F}(W)dt}}{{\cal
F}(W)(C+\int^{x}e^{-2\int H_{00}(t){\cal F}(W)dt}du)}\,.$ (42)

$C$ is an integration constant, which we assume to be real. Utilizing that
$H_{00}(x)={\cal F}^{-1}(x)(\ln U(x))^{\prime}$ the resulting deformation
coincides with Eq.(7). We thus therefore conclude that non-linear deformations
on the form Eq.(37) does not generate additional deformations to the ones
already generated by Eq.(1).

##  V Deforming the Coulomb potential

As a relatively simple application of the linear deformation scheme let us
briefly consider deformations of the Coulomb potential. This potential has,
within the framework of supersymmetric quantum mechanics, been treated in
several previous works Asim . The superpotential and the partner potential for
the Coulomb potential are given by Asim

$\displaystyle W(x)$ $\displaystyle=$
$\displaystyle\frac{q^{2}}{2(l+1)}-\frac{(l+1)}{x}\,,$ (43) $\displaystyle
V_{+}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\frac{q^{2}}{l+1})^{2}-\frac{q^{2}}{x}+\frac{(l+1)(l+2)}{x^{2}}\,.$
(44)

$q$ and $l$ in these expressions are the electric charge and the angular
momentum quantum numbers, respectively. These potentials result in the
following general solution for $U_{0}(x)$ in Eq.(9) Jensen

$U_{0}(x)=C_{1}M_{l+1,l+\frac{3}{2}}(\frac{q^{2}x}{l+1})+C_{2}W_{l+1,l+\frac{3}{2}}(\frac{q^{2}x}{l+1})\,.$
(45)

The $M(x)$\- and $W(x)$-functions are the Whittaker functions. The solution
Eq.(10) is given by Jensen

$U_{0}(x)\sim e^{\frac{q^{2}x}{2(l+1)}-(l+1)\ln(2x)}\,.$ (46)

We will for simplicity assume this solution in the following. We will let
$C_{01}\rightarrow\infty$ in Eq.(7) such that we deal with the identity
deformation $\hat{W}_{0}(x)=\hat{W}_{00}(x)=W(x)$. We will also ignore the
$Q_{j}(x)$ contributions in the following. Define $A\equiv q^{2}/(2(l+1))$ and
$B\equiv l+1$. It then follows that

$P_{1}(x)=C_{11}+C_{12}\int^{x}t^{2B}e^{-2At}dt\,,$ (47)

such that

$\hat{W}_{10}(x)=A-\frac{B}{x}+\frac{C_{12}x^{2B}e^{-2Ax}}{C_{11}+C_{12}\int^{x}t^{2B}e^{-2At}dt}\,.$
(48)

Let us consider the $s$-state with $l=0$ in order to get a better grasp on the
content buried in Eq.(48). We also set $q\equiv 1$. The expression for
$\hat{W}_{10}(x)$ then reduces to

$\hat{W}_{10}(x)=\frac{1}{2}-\frac{1}{x}+\frac{C_{12}x^{2}e^{-x}}{C_{11}-C_{12}(x^{2}+2x+2)e^{-x}}$
(49)

after redefining $C_{11}$ such that the lower integration limit of the
integral in Eq.(48) does not appear explicitly in the expression for the
potential. We will automatically do such redefinitions in the following when
it is appropriate. The corresponding physical potential $\hat{V}_{-1}(x)$ can
either be derived from the definition
$\hat{V}_{-1}(x)\equiv\hat{W}_{10}^{2}(x)-\hat{W}_{10}^{\prime}(x)$ or from
Eq.(11) with $\hat{W}_{00}(x)=W(x)$ and $C_{01}$ finite. This is a consequence
of a regeneration of Eq.(7) by the recursion scheme which was noted in Figure
1. From the definition it follows that

$\displaystyle\hat{V}_{-1}(x)=\frac{1}{4}-\frac{1}{x}+\frac{C_{12}x(2x-4)e^{-x}}{C_{11}-C_{12}(x^{2}+2x+2)e^{-x}}+\frac{2C_{12}^{2}x^{4}e^{-2x}}{(C_{11}-C_{12}(x^{2}+2x+2)e^{-x})^{2}}\,.$
(50)

In the special case when we set $C_{11}=0$ the last term in Eq.(49) becomes
independent of the exponentials (and $C_{12}$) and thus reduces to a pure
rational function. The physical potential $\hat{V}_{-1}(x)$ generated by
$\hat{W}_{10}(x)$ is then given by

$\hat{V}_{-1}(x)=\frac{1}{4}-\frac{1}{x}+\frac{4x(x+2)}{(x^{2}+2x+2)^{2}}\equiv
V_{-}(x)+\frac{4x(x+2)}{(x^{2}+2x+2)^{2}}\,.$ (51)

Let us go to the second iteration level starting from the expression for
$\hat{W}_{10}(x)$ in Eq.(49) with $C_{11}=0$, for convenience. It then follows
that

$\displaystyle\hat{W}_{20}(x)=\hat{W}_{1}(x)+\frac{C_{22}x^{2}e^{x}}{C_{21}(x^{2}+2x+2)^{2}+C_{22}(x^{2}+2x+2)e^{x}}\,.$
(52)

Note that when $C_{21}=0$ we get $\hat{W}_{20}(x)=W(x)$. Hence, the
deformation scheme allows in general for the possibility that additional
iterations in particular cases may regenerate previous potentials in a
nontrivial fashion. The expression for the corresponding physical potential is
given by

$\displaystyle\hat{V}_{-2}(x)=\hat{V}_{-1}(x)+\left[\frac{C_{22}x^{2}e^{x}}{C_{21}(x^{2}+2x+2)^{2}+C_{22}(x^{2}+2x+2)e^{x}}\right]\times$
$\displaystyle\times\left[-4(\frac{1}{x}+\frac{\frac{1}{2}x^{2}}{x^{2}+2x+2})+\frac{2(C_{21}(2x+2)+C_{22}e^{x})}{C_{21}(x^{2}+2x+2)+C_{22}e^{x}}\right]\,.$
(53)

The superpotential stemming from the third iteration with $\hat{W}_{20}(x)$ in
Eq.(52) as the starting point is given by

$\hat{W}_{30}(x)=\hat{W}_{20}(x)+\dfrac{\left(\dfrac{C_{32}x^{2}e^{x}}{C_{21}(x^{2}+2x+2)^{2}+C_{22}(x^{2}+2x+2)e^{x}}\right)}{\left(C_{31}+\dfrac{C_{32}}{C_{22}}\ln\left|\dfrac{C_{21}(x^{2}+2x+2)+C_{22}e^{x}}{x^{2}+2x+2}\right|\right)}\,.$
(54)

This superpotential introduces the possibility for a logarithmic singularity
away from the origin when $C_{22}/C_{21}<0$. We note that setting $C_{31}=0$
does not regenerate a previous potential as was possible at the previous
iteration level when we correspondingly put $C_{21}=0$. From Eq.(54) we can
deduce the physical potential $\hat{V}_{-3}(x)$ at the third iteration level.
We do not reproduce it here due to its complexity. Due to the complicated
integrals appearing we are not able to provide the analytical expression for
$\hat{W}_{40}(x)$. We leave detailed studies of the Coulomb potential for the
future.

##  VI Conclusion

In a previous paper we showed that isospectral deformations on the form Eq.(3)
are contained in the space of deformations generated by isospectral
deformations on the form Eq.(1). In this paper we have shown that Eq.(1) can
be considered as the initial, or base, deformation of a novel infinite
recursive isospectral deformation chain. This thus answers to some extend the
question by which we ended our previous paper Jensen ; how does the most
general isospectral deformation of the kind considered there (Eq.(1) in this
paper) look like. The results in this paper do obviously only give a partial
answer. We deduced in particular that a class of recursive deformations exists
which is generated by the solutions of the non-linear differential equations
in Eq.(16).

We briefly discussed various ways to construct alternative recursive
deformation structures. We considered a linear product structure, polynomial
deformations and completely generalized base deformations. They all either
failed to provide a recursive structure or they turned out to be identical to
the linear deformation scheme.

We applied the linear recursive scheme to the Coulomb potential. We derived
novel superpotentials which all satisfy the isospectrality condition. This
application did also demonstrate how easily novel isospectral deformations can
be generated in this approach. It did also demonstrate an increased relative
complexity of the generated potentials with the number of iterations, as one
also naively would expect from the expression Eq.(21).

## References

## References

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