# Quadratis Puzzles

Mario Gutiérrez1, Hugo Parlier2 and Paul Turner3
1Mixed Reality Group, Logitech Europe<EMAIL_ADDRESS>
2Department of Mathematics, University of Luxembourg<EMAIL_ADDRESS>
3Section de mathématiques, University of Geneva<EMAIL_ADDRESS>
###### Abstract

We introduce a family of reconfiguration puzzles arising from ideas in
geometry and topology. We present their construction from square-tiled shapes,
discuss some of the underlying mathematics and describe how they are naturally
associated to puzzle spaces which can be explored through visualization.

## Introduction

There is a historic relationship between reconfiguration puzzles, such as the
15 puzzle or the Rubik’s cube, and mathematics. In particular, there are many
so-called “sliding puzzles” where pieces slide around a board (see for
instance [4] for a collection of examples). In another direction, Weeks [11]
took standard puzzles and transferred them onto a torus.

In this paper we discuss a new family, Quadratis puzzles, which are inspired
by topological and geometric constructions of surfaces. These puzzles also
come with an associated “space” of possible configurations, that can be
studied and even visualized. In contrast, the so-called Rubik’s graph was
studied for decades, resulting in a computation of its diameter [6, 5], but
which is very far from being visualizable. The reader can also directly
experience the puzzles for themselves by playing them [10].

## The mathematics behind the puzzles

### Defining puzzles

To define a puzzle we must first describe what is meant by a board and a color
scheme. A board is constructed by pasting together (unit) squares: we assemble
collection of $N$ squares, each with upper, lower, left and right edges, by
pasting the edges together in such a way so that each left edge is pasted to a
right edge, and each upper edge is pasted to a lower edge. This results in an
orientable surface tiled by squares (see discussion below). In practice, we
often begin with a square-tiled shape that lives in the Euclidean plane, such
as the one shown in Figure 1.

(a) A square-tiled shape

(b) Left-right pasting

(c) Upper-lower pasting

Figure 1:

The non-pasted edges are then paired following the left-right or upper-lower
conventions. One possible pasting of Figure 1 is shown in Figure 1 (for left-
right paring) and 1 (for upper-lower).

Given a square-tiled shape, there is an “obvious” way to paste the remaining
sides together. Taking each horizontal line in turn, paste the first (ordering
from left to right) free left side to the first available free right side on
the same line. A similar procedure can be used to paste upper sides to lower
sides in the same column. We’ll refer to this pasting scheme as the standard
pasting. In Figure 2 and Figure 2 we see the standard pastings of two
different square-tiled shapes. Figure 1 and 1 present pasting that is not the
standard pasting.

(a)

(b)

Figure 2: Two square-tiled shapes and their standard pastings

The squares of a board will be colored from a fixed palette with a
predetermined number of squares of each given color. We refer to this
information as a color scheme. Formally, though this is unnecessary for this
article, we have a positive integer $K\leq N$ together with a $K$-tuple
$(n_{1},n_{2},\ldots,n_{K})$ such that $\sum_{n_{i}}=N$. Note that a color
scheme does not determine how to color each square of a board, only the number
of each color we must have.

Given a color scheme a particular coloring of a board (of which there are
many) will be referred to as a configuration. In Figure 3 we see five
different configurations of the board in Figure 2 with the color scheme
requiring 1 pink square, 4 blue squares and 4 purple squares.

Figure 3: Five of the possible $630$ configurations with the given color
scheme

To make a puzzle out of a board and a color scheme, we introduce a dynamic
element, via moves which transform one configuration into another. A single
square on a board always has $4$ (not necessarily distinct) neighboring
squares: the ones above, below, to the right and to the left. “Pushing” a
given square upwards, will have a knock-on effect on its upstairs neighbor,
pushing it upwards as well. This in turn pushes its neighbor and so on and so
forth until we end up back where we started. A coherent collection of squares
are all displaced by one step upwards. A similar thing is seen if we push
downwards, or to the left or to the right. We refer to these as moves. Figure
4 illustrates a left to right horizontal move involving all squares of the
middle row. This can be effected by “pushing” any square in that row towards
the right. The reader is encouraged to visit [10] to try this out for
themselves.

Figure 4: Making a move

Out of all of this we may now make a puzzle. Choose a particular configuration
as the home configuration: the aim will be to reproduce this configuration.
Starting with the home configuration we can apply a number of random moves (in
any direction) to obtain a new configuration. We refer to this as shuffling.
The goal is to put the squares back into the home configuration, by using the
moves described above. No distinction is made between two squares of the same
color: if two squares of the same color are swapped in the process of solving,
we do not care, the puzzle is still considered solved. Note that if a puzzle
was shuffled using $m$ moves, it is entirely possible that it can be solved
with fewer than $m$ moves. Pretty quickly, players tend to want to find the
optimal number of moves to solve a puzzle, which brings us to the main
mathematical object associated to a board equipped with a color scheme: its
puzzle space.

### The configuration spaces of puzzles

The puzzle space of a puzzle is a graph whose vertices are configurations and
where two configurations are related by an edge if they can be related by a
single move. We are only interested in configurations that can in principle be
solved, so we simply ignore the other configurations. This amounts to only
considering the component of the graph of all configurations which contains
the puzzle home configuration. Thus, by definition, a puzzle space is a
connected graph.

One could instead consider all possible colorings of a board with a prescribed
set of colors, and then relate configurations as above. The graph obtained
this way won’t always be connected, and in fact determining the number and
size of components seems to be an interesting problem. In [1], there is an
elementary proof of the fact that a standard 3 by 3 torus (see Figure 2 (a))
with all squares colored differently is disconnected.

In Figure 5 we see a simple example which can be worked out by hand. The board
is a two-by-two square with the standard pasting, and we color the squares
with two colors and two squares of each color. Any of the 6 configurations
shown may be used as the home configuration.

Figure 5: A simple puzzle space

As the complexity of the puzzle increases, the graphs become more complicated
and enjoy richer geometries. For instance, Figure 6 is the graph for a three-
by-three square board with standard pasting and two colors, 3 of one color and
6 of the other. As all configurations are related by moves in this example,
the choice of home configuration is again unimportant. The figure represents
one of many possible planar representations of the graph, and was obtained
using standard graph visualisation software (Gephi) and a force-directed graph
layout algorithm.

(a)

(b)

Figure 6: Puzzles spaces display intriguing features

### Relationships to moduli theory

By standard surface topology techniques the pasting procedure described above
results in an orientable surface equipped with a decomposition into squares.
In the examples shown in Figure 1 and Figure 2, the underlying surface is
connected, but that is not necessarily a requirement. Interestingly, one can
show that if one randomly pastes the squares together, the result is often
connected. In fact, the probability of obtaining a connected surface goes to
$1$ as the number of squares goes to infinity [9, 8].

The board given by Figure 2 is a genus $2$ surface (so topologically a double-
torus). This can be shown by a standard Euler characteristic computation and
the classification of surfaces. By using the Euler characteristic again, it
can be proved that the board from Figure 1 is a genus $6$ surface.

Beyond the topology and the combinatorics of the decomposition into squares,
the boards are surfaces with a geometric structure. In fact, they are
important objects in the theory of moduli spaces, and are generally called
square-tiled translation surfaces. (The word “translation” comes from the fact
that side identifications are made by Euclidean translations.) These surfaces
have all sorts of interesting properties, including being dense in the space
of all translation surfaces (up to scaling). Some of them are quite exotic and
have fascinating dynamical and geometric properties [7] and there has been a
lot of research about square-tiled and translation surfaces in general, a
partial survey can be found in [3]. As an example, a very natural question is:
how many square-tiled translation surfaces can be made with $N$ squares? It
turns out to be very difficult and is related to computing volumes of moduli
space (see [2] for partial results). In general the answer is not known,
making the question of enumerating the number of possible puzzle boards with a
given number of squares, an open and difficult problem.

For these reasons, the puzzle spaces we introduce above can be seen as “toy”
moduli spaces, and can be studied using a similar approach. How many vertices
are there? How do you determine the distance between two configurations? More
generally, what do these spaces “look” like?

These questions, as for moduli spaces, can range from specific to more
general, from elementary to more sophisticated. For instance, what are their
diameters? Whereas a very limited number of small examples can be worked out,
finding a general answer to this type of question very quickly turns into
serious and seemingly difficult research questions.

## Visualizing puzzle spaces

There are two desirable properties we might ask of a puzzle, namely that it is
interesting and visualizable. These are rather loose concepts: interesting
means non-trivial yet doable in the sense that a player can, with some
thought, develop strategies which help to provide a solution; visualizable
means that the puzzle space is of a size that may be displayed so that the
vertices and edges remain discernible. These two constraints pull in different
directions with increasing interest often resulting in graphs that are too
large.

### The first examples: chroma-squares

A particularly simple class of puzzles was introduced in [1] under the name
chroma-squares: the board is a square $N\times N$ array and the color scheme
has $N$ colors with $N$ squares of each. The pasting is the standard one,
meaning the underlying surface is a torus.

Figure 7: Five of the approximately $2.7\times 10^{24}$ configurations of size
$6\times 6$, images from [1]

There are some rather obvious choices to make for home configurations, namely
lines or columns of uniform color. With essentially no mathematical background
it is possible to explain that any two configurations can be linked by a
sequence of moves (see [1]), which shows that the graph of all configurations
is connected.

With 1680 vertices and around 10,000 edges, the case $N=3$ yields a puzzle
that is both interesting and visualizable. For $N=2$ the puzzle is too trivial
and the graph too simple (it is given in Figure 5 above) and for $N>3$ the
graph is too large (although the difficultly of solving the puzzle does not
really increase).

### Beyond tori

As we saw above, the underlying surface of a puzzle is an orientable surface.
Higher genus examples also give a rich source of interesting and visualizable
puzzle spaces. Recall that the board shown in Figure 2 has genus two. With
color scheme indicated in Figure 3 it is a relatively easy puzzle (for any
chosen home configuration), but not completely trivial, and it possesses a
visualizable puzzle graph as shown in Figure 6.

### Exploring large puzzles

While many puzzles are both interesting and visualizable, sometimes we may
want to partially visualize a configuration space of a puzzle complicated
enough so the entire space is beyond visualization (either for computational
or presentational reasons). The puzzle shown in Figure 8 (in which the board
is a $5\times 5$ grid missing its middle square) is interesting because the
usual strategies do not resolve a potential problem of switching at the end.
There is something new here, which means it is worth retaining as an example
despite its large configuration space (which has has over 3000 billion billion
points, roughly the number of square millimeters needed to cover the entirety
of the Grand Duchy of Luxembourg).

(a) Exploratis installation at the Luxembourg Science Center

(b) A puzzle which is too big to be explored entirely

Figure 8:

One solution is to dynamically develop the graph as the player walks through
different configurations: if a new configuration is reached, add it to the
graph along with the edge used to reach it. This is the basis of the Science
Museum installation Exploratis developed in conjunction with the Luxembourg
Science Center in which an iPad driven puzzle is linked to a large screen
displaying an ever larger portion of the configuration space (see Figure 8).
Here again, a type of repulsion based algorithm is used to visualize the
graph. Players can play individually, race or collaborate. This time, because
the graph changes with every move, the visuals have a wow-effect.

(a)

(b)

(c)

Figure 9:  Exploratis installation at Expo Dubai (Luxembourg Pavilion)

A similar installation was made available to visitors of the Luxembourg
Pavilion at the World Expo in Dubai. Hundreds of visitors each day were able
to test the puzzles and explore the graphs. In each case, the puzzles are
tailored for the event. While many of the puzzles that we use are the same as
the one in the available app, we can create new puzzles, such as one
representing the flag of Luxembourg for the pavilion, for special
installations.

## Conclusion

Although only recently made public, we have already had several opportunities
to test the Quadratis puzzles with a variety of different player profiles, and
we get continual feedback from the Science Center. Something that was not
obvious to us at first, and that required a lot of testing, is that the
puzzles can be scaled in difficulty, ranging from completely obvious to
(apparently) beyond the reach of human capability (and with basically
everything in between). By careful planning, a sequence of puzzles can be
presented that allows players of any level to learn as they progress. The
lessons learnt, along with tricks and techniques uncovered through play, can
be subsequently applied to increasingly challenging situations.

So why is this interesting? The puzzles can be appreciated without any
scientific background, and yet the surrounding mathematics gives a player a
glimpse at what it is like to explore unchartered mathematical territory. In
light of the feedback we’ve received so far, we see this is a starting point
for a multi-faceted adventure. What is particularly exciting is to see the
diversity of players (background, origin, age…) that have experienced the joy
of simply playing these puzzles. As we create more and more puzzles, we are
ourselves discovering new and interesting phenomena. The exploration of the
mathematics, particularly the study of the associated graphs, has only just
started. In a different direction, it would be fascinating to know more about
what and how players learn while progressing though a sequence of puzzles.

Quadratis puzzles are regarded as simple fun by some, but the fact that they
can be used to illustrate mathematics and the nature of mathematical research
remains, for us, a central motivation.

## Acknowledgements

This article is part of a more general long standing project aiming to
illustrate mathematical research in different ways, and we’ve received a lot
of support along the way.

The original Mathema project was co-financed by the Swiss National Science
Foundation and by TomBooks. In particular, we thank Thomas Steinmann for his
personal support.

The exhibitions where we have presented Quadratis and Exploratis are brought
alive by mediators so many thanks to all of you who have participated in
science festivals. There are too many of you to name, but a particular thank
you goes to Bruno Teheux. Not only was he co-organiser of many of the events
but his feedback and encouragement has been invaluable. More generally, the
University of Luxembourg, and in particular the Department of Mathematics of
the Faculty of Science, Technology and Medicine, has provided logistical and
financial aid. Thank you in particular to Giovanni Peccati. We also thank
Sorbonne Abu Dhabi and the University of Geneva for help with certain
exhibitions and to the Luxembourg Pavillion at the Expo in Dubai for hosting
us.

The Exploratis station had the support of many partners. First of all, thank
you to the Luxembourg Science Center for hosting us, and to Julien Meyer for
his help in making this vision a reality. Exploratis was supported by a PSP
Classic grant from the Luxembourg National Science Foundation, who has also
supported the project indirectly in many ways (Researchers’ Days and Science
Festivals). Exploratis was also co-developed with Reyna Juárez who went far
and beyond the call of duty.

## References

  * [1] Hugo Parlier and Paul Turner, Mathema, TomBooks, 2015.
  * [2] A. Eskin and A. Okounkov. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145 (2001), no. 1, 59–103.
  * [3] G. Forni and C. Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. J. Mod. Dyn. 8 (2014), no. 3–4, 271–436.
  * [4] E. Hordern. Sliding Piece Puzzles. Oxford University Press, 1986.
  * [5] E. Demaine, M. Demaine, S. Eisenstat, A. Lubiw, A.Winslow. Algorithms for solving Rubik’s cubes. Algorithms – ESA 2011, 689–700, Lecture Notes in Comput. Sci., 6942, Springer, Heidelberg, 2011.
  * [6] T. Rokicki, H. Kociemba, M. Davidson and J. Dethridge. The diameter of the Rubik’s cube group is twenty. SIAM J. Discrete Math. 27 (2013), no. 2, 1082–1105.
  * [7] G. Schmithüsen. Examples for Veech groups of origamis. The geometry of Riemann surfaces and abelian varieties, 193–206, Contemp. Math., 397, Amer. Math. Soc., Providence, RI, 2006.
  * [8] S. Shrestha, J. Wang. The topology and geometry of random square-tiled surfaces, arxiv preprint, 2020.
  * [9] S. Shrestha, J. Wang. Statistics of Square-tiled Surfaces: Symmetry and Short Loops, arxiv preprint, 2019.
  * [10] The website quadratis.app with puzzles and links to downloads.
  * [11] J. Weeks. Torus games. Available on the AppStore, 2011.


# One continuous parameter family of Dirac Lorentz scalar potentials
associated with exceptional orthogonal polynomials

Suman Banerjeea111e-mail address: suman.raghunathpur@gmail.com(S.B) and Rajesh
Kumar Yadava222e-mail address: rajeshastrophysics@gmail.com(R.K.Y)

###### Abstract

We extend our recent works [Int. J. Mod. Phys. A 38 (2023) 2350069-1] and
obtain one parameter $(\lambda)$ family of rationally extended Dirac Lorentz
scalar potentials with their explicit solutions in terms of $X_{m}$
exceptional orthogonal polynomials. We further show that as the parameter
$\lambda\rightarrow 0$ or $-1$, we get the corresponding rationally extended
Pursey and the rationally extended Abraham-Moses type of scalar potentials
respectively, which have one bound state less than the starting scalar
potentials.

$\leavevmode\nobreak\ {}^{a}$Department of Physics, Sido Kanhu Murmu
University, Dumka-814110, India.

##  1 Introduction

In non-relativistic quantum mechanics, after the discovery of
$X_{m}$-exceptional orthogonal polynomials (EOPs) [1, 2, 3], several new
potentials have been discovered whose bound state solutions are in terms of
these EOPs [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. These
potentials are generally the rational extension of the known conventional
potentials and hence also known as rationally extended (RE) potentials. Out of
these RE potentials some are exactly solvable and few are quasi exactly or
conditionally exactly solvable potentials. Different approaches such as the
Darboux-Crum-transformation (DBT), Point Canonical transformation (PCT) and
Supersymmetry in quantum mechanics (SQM) etc. have been adopted to obtain
these potentials and their solutions.

One obvious question is whether there are relativistic problems whose solution
is also in terms of these EOPs. As a first step in that direction, recently we
[20] considered Dirac equation with Lorentz scalar potential and showed that
indeed the exact eigenfunctions for some of the Lorentz scalar potentials are
in terms of these EOPs. In particular, we showed that the exact solutions for
Dirac potentials (with Lorentz scalar coupling) corresponding to RE radial
oscillator potential, trigonometric scarf potential and generalized Pöschl-
Teller (GPT) potential can be obtained in terms of these EOPs. One related
question is, are there strictly isospectral RE Dirac potentials corresponding
to these three Dirac potentials? Further, are there corresponding RE Pursey
and Abraham Moses (AM) potentials with one bound state less? The purpose of
this note is to answer these questions in the affirmative. In particular,
starting from these three RE Dirac scalar potentials and using the formalism
of SQM, we construct one continuous ($\lambda$) parameter family of strictly
isospectral Dirac potentials. Further, by taking the limit of $\lambda=0$ and
-$1$, we obtain the corresponding RE Pursey and RE AM potentials.

The plan of the paper is as follows. In section $2$, we briefly discuss the
general formalism explaining how using SQM formalism one can construct one
continuous parameter family of strictly isospectral Dirac Lorentz scalar
potentials. We also indicate how to construct the corresponding Pursey and AM
potentials with one bound state less. In section 3, we apply the above
formalism to the case of the RE radial oscillator potential and obtain the
corresponding one parameter family of strictly isospectral Dirac Lorentz
scalar potentials as well as the corresponding RE Pursey and RE AM potentials.
In order to avoid unnecessary details, the results associated with the RE
Scarf and RE GPT potentials are summarized in Tables $1$, $2$ and $3$.
Finally, we summarize the results in section 4.

##  2 Formalism

To set up the notations, let us consider one dimensional Dirac equation with a
Lorentz scalar potential $\phi(x)$ given by [21]

$[i\gamma^{\mu}\partial_{\mu}-\phi(x)]\Psi_{n}(x,t)=0,\quad\mu=0,1$ (1)

where $\Psi_{n}(x,t)$ is Dirac spinor defined in the matrix form as

$\Psi_{n}(x,t)=\begin{bmatrix}\Psi^{(1)}_{n}(x,t)\\\ \Psi^{(2)}_{n}(x,t)\\\
\end{bmatrix}$ (2)

If we put $\Psi_{n}(x,t)=\exp(-i\epsilon t)\Psi_{n}(x)$ (where $\epsilon=$
energy associated with the spectrum) the above Dirac equation Eq. (1) takes
the form

$\gamma^{0}\epsilon\Psi_{n}(x)+i\gamma^{1}\frac{d}{dx}\Psi_{n}(x)-\phi_{n}(x)\Psi_{n}(x)=0\,.$
(3)

We choose a $2D$ representation of the gamma matrices [25] to directly cast
the problem in one dimensional SUSY form i.e,

$\gamma^{0}=\sigma_{x}=\begin{bmatrix}0&1\\\ 1&0\\\
\end{bmatrix},\qquad\gamma^{1}=i\sigma_{z}=\begin{bmatrix}i&0\\\ 0&-i\\\
\end{bmatrix}$ (4)

and

$\Psi_{n}(x)=\begin{bmatrix}\Psi^{(1)}_{n}(x)\\\ \Psi^{(2)}_{n}(x)\\\
\end{bmatrix},$ (5)

so that we have two coupled equations corresponding to the Dirac equation (3)
given by

$\hat{A}\Psi^{(1)}_{n}(x)=\epsilon\Psi^{(2)}_{n}(x)\quad\mbox{and}\quad\hat{A^{\dagger}}\Psi^{(2)}_{n}(x)=\epsilon\Psi^{(1)}_{n}(x)\,,$
(6)

where the operators,

$\hat{A}=\frac{d}{dx}+\phi(x)\quad\mbox{and}\quad\hat{A^{\dagger}}=-\frac{d}{dx}+\phi(x)\,.$
(7)

Now the above Eq.(6) can be decoupled easily (in terms of
$H_{1}(=\hat{A}\hat{A}^{\dagger})$ and
$H_{2}(=\hat{A^{\dagger}}\hat{A})$) and written as

$H_{2}\Psi^{(1)}_{n}(x)=\epsilon^{2}\Psi^{(1)}_{n}(x)\quad\mbox{and}\quad
H_{1}\Psi^{(2)}_{n}(x)=\epsilon^{2}\Psi^{(2)}_{n}(x)\,,$ (8)

which are equivalent to two Schrödinger like equations namely

$-\frac{d^{2}}{dx^{2}}\Psi^{(1)}_{n}(x)+V^{(1)}(x)\Psi^{(1)}_{n}(x)=\epsilon^{2}\Psi^{(1)}_{n}(x)\,,$
(9)
$-\frac{d^{2}}{dx^{2}}\Psi^{(2)}_{n}(x)+V^{(2)}(x)\Psi^{(2)}_{n}(x)=\epsilon^{2}\Psi^{(2)}_{n}(x)\,,$
(10)

with potential like terms

$V^{(1)}(x)=\phi^{2}(x)-\phi^{{}^{\prime}}(x)\,,$ (11)

and

$V^{(2)}(x)=\phi^{2}(x)+\phi^{{}^{\prime}}(x)\,.$ (12)

On comparing with the well known formalism of SQM [22], we see that there is
supersymmetry in the problem and the scalar potential $\phi(x)$ is just the
superpotential of the SQM formalism and $V^{(1,2)}$ being the partner
potentials. Thus the eigenvalues and the eigenfunctions of the two
Hamiltonians $H_{1}$ and $H_{2}$ are related except that one of them has an
extra bound state at zero energy so long as $\phi(x\rightarrow\pm\infty)$ have
opposite signs. Without any loss of generality we always choose $\phi(x)$ such
that the ground state energy of $H_{1}$ is zero. In that case the
eigenfunctions and the eigenvalues of the two Hamiltonians are related as
follows

$\Psi_{n}^{(2)}(x)=[E_{n+1}^{(1)}]^{-1/2}A\Psi_{n+1}^{(1)}(x)\,,$ (13)
$\Psi_{n+1}^{(1)}(x)=[E_{n}^{(2)}]^{-1/2}A^{+}\Psi_{n}^{(2)}(x)\,,$ (14)
$E_{n}^{(2)}=E_{n+1}^{(1)}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
E_{0}^{(1)}=0\,,$ (15)

while the scalar potential $\phi(x)$ is related to the zero energy ground
state eigenfunction by

$\phi(x)=-\frac{d}{dx}[\ln\Psi_{0}^{(1)}(x)]$ (16)

Finally, it is well known [22, 23, 24] that given any potential $V^{(1)}(x)$
with at least one bound state, it is straight forward to construct one
continuous parameter family of scalar potentials $\phi(x,\lambda)$ (which are
nothing but the super potentials in the present case) and hence one continuous
parameter family of potentials $V^{(1)}(x,\lambda)$. Here $\lambda$ can take
any value $>0$ or $<-1$. In particular, the corresponding one parameter family
of scalar potentials $\phi(x,\lambda)$ are given by [23]

$\phi(x,\lambda)=\phi(x)+\frac{d}{dx}\ln[I(x)+\lambda]\,,$ (17)

where $I(x)$ in terms of the normalized ground state eigenfunction
$\Psi_{0}^{(1)}(x)$ is given by

$I(x)=\int_{-\infty}^{x}[\Psi_{0}^{(1)}(y)]^{2}\,dy\,.$ (18)

The corresponding one continuous parameter family of strictly isospectral
potentials are

$V^{(1)}(x,\lambda)=V^{(1)}(x)-2\frac{d^{2}}{dx^{2}}\ln[I(x)+\lambda]\,,$ (19)

with the corresponding normalized ground state eigenfunctions being

$\hat{\Psi}^{(1)}_{0}(\lambda,x)=\frac{\sqrt{\lambda(1+\lambda)}\Psi^{(1)}_{0}(x)}{\big{(}I(x)+\lambda\big{)}}$
(20)

Thus the ground state wave functions corresponding to Eq.(17) can be expressed
as

$\hat{\Psi}_{0}(\lambda,x)=\begin{bmatrix}\hat{\Psi}^{(1)}_{0}(\lambda,x)\\\
0\\\ \end{bmatrix}$ (21)

The normalized excited-state $(n=0,1,2....)$ eigenfunctions are obtained as-

$\hat{\Psi}_{n+1}(\lambda,x)=\begin{bmatrix}\hat{\Psi}^{(1)}_{n+1}(\lambda,x)\\\
\Psi^{(2)}_{n}(x)\\\ \end{bmatrix}\sqsupset$ (22)

where,

$\hat{\Psi}^{(1)}_{n+1}(\lambda,x)=\Psi^{(1)}_{n+1}(x)+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I^{\prime}(x)}{I(x)+\lambda}\bigg{)}\bigg{(}\frac{d}{dx}+\phi(x)\bigg{)}\Psi^{(1)}_{n+1}(x).$
(23)

Finally, as $\lambda\rightarrow 0$ and $-1$, we get the Pursey and the AM like
scalar potentials respectively which are isospectral to $V^{(2)}(x)$. We
summarize the key features of the two potentials as below:

(a) The Pursey potential:

For $\lambda=0$, we get Pursey like Dirac potential from Eq.(17),

$\hat{\phi}(\lambda=0,x)=\phi^{[P]}(x)=\phi(x)+\frac{d}{dx}\ln[I(x)]\,.$ (24)

The corresponding eigenfunctions are

$\displaystyle\Psi^{[P]}_{n}(x)$ $\displaystyle=$
$\displaystyle\hat{\Psi}^{(1)}_{n+1}(\lambda=0,x)$ (25) $\displaystyle=$
$\displaystyle\Psi^{(1)}_{n+1}(x)+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I^{\prime}(x)}{I(x)}\bigg{)}\bigg{(}\frac{d}{dx}+\phi(x)\bigg{)}\Psi^{(1)}_{n+1}(x),$

while the energy eigen value for the $\lambda=0$ are

$E^{[P]}_{n}=E^{(2)}_{n}.$ (26)

(b) The Abraham-Moses potential:

For $\lambda=-1$, we get AM like Dirac potential from Eq.(17),

$\hat{\phi}(\lambda=-1,x)=\phi^{[AM]}(x)=\phi(x)+\frac{d}{dx}\ln[I(x)-1]\,.$
(27)

The corresponding eigenfunctions are

$\displaystyle\Psi^{[AM]}_{n}(x)$ $\displaystyle=$
$\displaystyle\hat{\Psi}^{(1)}_{n+1}(\lambda=-1,x)$ $\displaystyle=$
$\displaystyle\Psi^{(1)}_{n+1}(x)+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I^{\prime}(x)}{I(x)-1}\bigg{)}\bigg{(}\frac{d}{dx}+\phi(x)\bigg{)}\Psi^{(1)}_{n+1}(x).$

while the energy eigen values for $\lambda=-1$ are

$E^{[AM]}_{n}=E^{(2)}_{n}\ .$ (29)

Proceeding in this way, for a system with $n$ bound states, one can also
obtain n-parameter family of Lorentz scalar potential by iterating the same
procedure $n$ times.

##  3 Examples

We now consider three different RE scalar potentials namely the RE radial
oscillator, RE Scarf-I and RE generalized Pöschl-Teller potentials and obtain
one-parameter family of corresponding Lorentz scalar potentials with their
solutions in terms of $X_{m}$ exceptional Orthogonal Polynomials. We discuss,
the example of radial oscillator case in detail and summarize the results for
the other two cases in Tables $1,2$ and $3$.

###  3.1 RE Radial oscillator type Lorentz scalar potential

In this case, we consider the $\phi(x)\rightarrow\phi_{m,ext}(r,\omega,\ell)$
given by [20]

$\phi_{m,ext}(r,\omega,\ell)=\phi_{con}(r,\omega,\ell)+\phi_{m,rat}(r,\omega,\ell);\quad
0\leq r\leq\infty$ (30)

where

$\phi_{con}(r,\omega,\ell)=\frac{\omega r}{2}-\frac{(\ell+1)}{r};\quad\ell>0$
(31)

and

$\phi_{m,rat}(r,\omega,\ell)=\omega
r\bigg{[}\frac{L^{(\alpha)}_{m-1}(-z)}{L^{(\alpha-1)}_{m}(-z)}-\frac{L^{(\alpha+1)}_{m-1}(-z)}{L^{(\alpha)}_{m}(-z)}\bigg{]},$
(32)

are the conventional and rational terms respectively. Here,
$L^{(\alpha)}_{m}(z)$ is the Laguerre polynomial, $z=\frac{\omega r^{2}}{2}$
and $\alpha=l+\frac{1}{2}$.

If we use this $\phi_{m,ext}(r,\omega,\ell)$ in equation (11), we get the
Schrödinger like equation (9), where

$V^{(1)}(x)\rightarrow
V^{(1)}_{m,ext}(r,\omega,\ell)=V_{con}(r,\omega,\ell)+V_{rat,m}(r,\omega,\ell)$
(33)

with terms

$V_{con}(r,\omega,\ell)=\frac{1}{4}\omega^{2}r^{2}+\frac{\ell(\ell+1)}{r^{2}}-\omega(\ell+\frac{3}{2})$
(34)

and

$\displaystyle V_{m,rat}(r,\omega,\ell)$ $\displaystyle=$
$\displaystyle-\omega^{2}r^{2}\frac{L^{(\alpha+1)}_{m-2}(-z)}{L_{m}^{(\alpha-1)}(-z)}+2\omega(z+\alpha-1)\frac{L^{(\alpha)}_{m-1}(-z)}{L_{m}^{(\alpha-1)}(-z)}$
(35) $\displaystyle+$ $\displaystyle
2\omega^{2}r^{2}\bigg{(}\frac{L^{(\alpha)}_{m-1}(-z)}{L_{m}^{(\alpha-1)}(-z)}\bigg{)}^{2}-2m\omega,\quad
0<r<\infty$

which is a well-known rationally extended radial oscillator potential. The
solution of the Eq. (9) in terms of $X_{m}$-exceptional Laguerre polynomials
$\hat{L}_{n+m}^{(\alpha)}(z)$ is thus given by

$\Psi^{(1)}_{n}(x)\rightarrow\Psi^{(1)}_{n,m,ext}(r,\omega,\ell)=N^{(\alpha)}_{n,m}f_{m}(\alpha,z)\hat{L}_{n+m}^{(\alpha)}(z),$
(36)

where,
$f_{m}(\alpha,z)=\frac{r^{\alpha+\frac{1}{2}}\exp\big{(}-\frac{z}{2}\big{)}}{L_{m}^{(\alpha-1)}(-z)}$
and the normalization constant

$N^{(\alpha)}_{n,m}=\bigg{[}\frac{n!\omega^{(\alpha+1)}}{2^{\alpha}(\alpha+n+m)\Gamma(\alpha+n)}\bigg{]}^{1/2}$
(37)

for $m=1,2,...$ and $n=0,1,2..,$. In terms of the classical Laguerre
polynomials the expression of $\hat{L}_{m}^{(\alpha)}(z)$ is given as [24]

$\hat{L}^{(\alpha)}_{n+m}(z)=L^{(\alpha)}_{m}(-z)L^{(\alpha-1)}_{n}(z)+L^{(\alpha-1)}_{m}(-z)L^{(\alpha)}_{n-1}(z);\quad
n\geq m.$ (38)

If we compare Eqs. (11) and (12), we observe that the $V^{(2)}(x)\rightarrow
V^{(2)}_{m,ext}(r,\omega,\ell)$ is the partner potential of
$V^{(1)}_{m,ext}(r,\omega,\ell)$ and hence the second component of the
eigenfunction $\Psi_{n}(x)\rightarrow\Psi_{n,m,ext}(r,\omega,\ell)$ i.e,
$\Psi^{(2)}_{n}(x)\rightarrow\Psi^{(2)}_{n,m,ext}(r,\omega,\ell)$ can easily
be obtained using Eq. (6) i.e,

$\Psi^{(2)}_{n,m,ext}(r,\omega,\ell)=N^{(\alpha+1)}_{n,m}f_{m}(\alpha+1,z)\hat{L}_{n+m}^{(\alpha+1)}(z).$
(39)

The energy eigenvalues are

$E^{(1)}_{n+1}=E^{(2)}_{n}=2(n+1)\omega\,,\leavevmode\nobreak\
\leavevmode\nobreak\ E^{(1)}_{0}=0\,.$ (40)

Here $n=0$ corresponds to the ground state solutions. Using the first
component of the ground state Dirac eigenfunction and following Eqs.(18) and
(17), one can obtain the integral $I(x)\rightarrow I_{m}(r,\omega,\ell)$ and
hence get the one parameter isospectral family of Lorentz scalar potentials
$\hat{\phi}_{m,ext}(\lambda,r,\omega,\ell)$ for given values of $m$ and
$\lambda$.

####  3.1.1 Illustration for $m=1$

As an illustration, we consider the $X_{1}$ case $(m=1)$ for which the
expression for the extended scalar potential (30) looks like

$\displaystyle\phi_{1,ext}(r,\omega,\ell)$ $\displaystyle=$
$\displaystyle\phi_{con}(r,\omega,\ell)+\phi_{1,rat}(r,\omega,\ell)$
$\displaystyle=$ $\displaystyle\frac{\omega
r}{2}-\frac{(\ell+1)}{r}+\frac{4\omega r}{(1+2\ell+\omega
r^{2})(3+2\ell+\omega r^{2})}$

The ground state Dirac eigenfunction is given by

$\Psi^{(1)}_{0,1,ext}(r,\omega,\ell)=N^{(\alpha)}_{0,1}\Bigg{(}\frac{3+2\ell+\omega
r^{2}}{1+2\ell+\omega r^{2}}\Bigg{)}r^{(1+\ell)}\exp{(-\frac{\omega
r^{2}}{4})}\,.$ (42)

Now to get the explicit expression for $I_{1}(r,\omega,\ell)$, we fix the
particular values of potential parameters (say $\omega=3$ and $\ell=1$) and
get

$I_{1}(r,3,1)=-\frac{{e^{-\frac{{3r^{2}}}{2}}\sqrt{\frac{{6}}{{\pi}}}r\left(5+10r^{2}+3r^{4}\right)}}{{5\left(1+r^{2}\right)}}+\operatorname{{erf}}\left(\sqrt{\frac{{3}}{{2}}}r\right),$
(43)

where the $\operatorname{{erf}}(r)$ is the error function. Thus, the
expression for the one parameter family (17) of rational radial oscillator
type scalar potentials are given by

$\displaystyle\hat{\phi}_{1,ext}(\lambda,r,3,1)$ $\displaystyle=$
$\displaystyle-\frac{\zeta(r)+\xi(r)(\lambda+\operatorname{{erf}}\left(\sqrt{\frac{3}{2}}r\right))}{{2r\left(\vartheta(r)-\Upsilon(r)(\lambda+\operatorname{{erf}}\left(\sqrt{\frac{3}{2}}r\right))\right)}},$
$\displaystyle\mbox{where,}\quad\zeta(r)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{6}{\pi}}r(100+145r^{2}+195r^{4}+117r^{6}+27r^{8}),$
$\displaystyle\xi(r)$ $\displaystyle=$ $\displaystyle
5e^{\frac{3r^{2}}{2}}(-20-9r^{2}+12r^{4}+9r^{6}),$ $\displaystyle\vartheta(r)$
$\displaystyle=$
$\displaystyle\sqrt{\frac{6}{\pi}}r(5+3r^{2})(5+10r^{2}+3r^{4})$
$\displaystyle\mbox{and}\quad\Upsilon(r)$ $\displaystyle=$ $\displaystyle
5e^{\frac{3r^{2}}{2}}(5+8r^{2}+3r^{4}).$ (44)

The corresponding normalized ground state eigenfunction has the form

$\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,r,3,1)=\frac{\sqrt{\lambda(1+\lambda)}\Psi^{(1)}_{0,1,ext}(r,3,1)}{[I_{1}(r,3,1)+\lambda]},$
(45)

where $\Psi^{(1)}_{0,1,ext}(r,3,1)$ and $I_{1}(r,3,1)$ are given by Eqs. (42)
and (43) respectively. One can also obtain the normalized excited-state
eigenfunctions

$\hat{\Psi}^{(1)}_{n,1,ext}(\lambda,r,3,1)=\Psi^{(1)}_{n+1}(r,3,1)+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I_{1}^{\prime}(r,3,1)}{I_{1}(r,3,1)+\lambda}\bigg{)}\bigg{(}\frac{d}{dr}+\phi_{1,ext}(r,3,1)\bigg{)}\Psi^{(1)}_{n+1,ext}(r,3,1).$
(46)

For $\lambda=0$, we get the RE Pursey type Lorentz scalar potentials i.e,

$\displaystyle\phi^{[P]}_{1,ext}(r)=-\frac{\zeta(r)+\xi(r)(\operatorname{{erf}}\left(\sqrt{\frac{3}{2}}r\right))}{{2r\left(\vartheta(r)-\Upsilon(r)(\operatorname{{erf}}\left(\sqrt{\frac{3}{2}}r\right))\right)}}$

with the corresponding eigenfunctions being

$\displaystyle\Psi^{[P]}_{n,1,ext}(r,3,1)=\bigg{[}1+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I_{1}^{\prime}(r,3,1)}{I_{1}(r,3,1)}\bigg{)}\bigg{(}\frac{d}{dr}+\phi_{1,ext}(r,3,1)\bigg{)}\bigg{]}\Psi^{(1)}_{n+1,1,ext}(r,3,1)$
(48)

while using Eq.(26) and (40) the energy eigenvalues are given by

$E^{[P]}_{n}=6(n+1);.$ (49)

Similarly, for $\lambda=-1$, we get the RE AM type Lorentz scalar potentials

$\phi^{[AM]}_{1,ext}(r,3,1)=-\frac{\zeta(r)+\xi(r)(-1+\operatorname{{erf}}\left(\sqrt{\frac{3}{2}}r\right))}{{2r\left(\vartheta(r)-\Upsilon(r)(-1+\operatorname{{erf}}\left(\sqrt{\frac{3}{2}}r\right))\right)}}\\\
$

with the corresponding eigenfunctions being

$\displaystyle\Psi^{[AM]}_{n,1,ext}(r,3,1)=\bigg{[}1+\frac{1}{E^{(1)}_{n+1}}\bigg{(}\frac{I_{1}^{\prime}(r,3,1)}{I_{1}(r,3,1)-1}\bigg{)}\bigg{(}\frac{d}{dr}+\phi_{1,ext}(r,3,1)\bigg{)}\bigg{]}\Psi^{(1)}_{n+1,1,ext}(r,3,1).$
(50)

while the energy eigenvalues are

$E^{[AM]}_{n}=E^{[P]}_{n}=6(n+1).$ (51)

The plots of $\hat{\Phi}_{1,ext}(\lambda,r,3,1)$, $\Phi^{[P]}(r,3,1)$,
$\Phi^{[AM]}(r,3,1)$ and the normalized ground state eigenfunctions
$\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,r,3,1)$ are shown in fig. $1(a)$ to $1(c)$
for different values of $\lambda$.

Fig.1: (a) Plot of $\hat{\phi}_{1,ext}(\lambda,r,3,1)$ vs. $r$ for positive
$\lambda(0,0.05,0.1,1$ and $\infty)$. REP type scalar potential is shown for
$\lambda=0$.

Fig.1: (b) Plots of $\hat{\phi}_{1,ext}(\lambda,r,3,1)$ for negative
$\lambda(-\infty,-1.1,-1.01,-1.001$ and $-1)$. The REAM type scalar potential
is shown for $\lambda=-1$.

Fig.1: (c) Normalized ground-state wavefunctions
$\frac{\hat{\Psi}^{(1)}_{0,1}(\lambda,r,3,1)}{r}$ for some potentials (with
positive $\lambda$).

In Fig.1(a), we observe that as $\lambda$ starts decreasing from $+\infty$ to
$0$, the Dirac potential starts developing maxima and the corresponding peaks
shifts towards $r=0$. On the other hand in Fig.1(b), as $\lambda$ increases
from $-\infty$ to $-1$, the depth of the curves are gradually decreasing and
these become flat as $\lambda\rightarrow-\infty$.

###  3.2 RE Scarf-I and RE GPT type Lorentz scalar potentials

Similar to the RE radial oscillator scalar potential case, the general form
for the RE Scarf-I and the RE GPT type Lorentz scalar potentials with their
solutions in terms of $X_{m}$ Jacobi polynomials333In terms of classical
Jacobi polynomials, the $X_{m}$-Jacobi is written as
$\hat{P}_{n+m}^{(\alpha,\beta)}(z)=(-1)^{m}\bigg{[}\frac{1+\alpha+\beta+n}{2(1+\alpha+n)}(z-1)P^{(-\alpha-1,\beta-1)}_{m}(z)P^{(\alpha+2,\beta)}_{n-1}(z)+\frac{1+\alpha-m}{\alpha+1+n}P^{(-2-\alpha,\beta)}_{m}(z)P^{(\alpha+1,\beta-1)}_{n}(z)\bigg{]}$.
and the corresponding energy eigenvalues are summarized in a tabular form and
shown in Table-$1$. Using these we can find the expressions for
$\phi_{m,ext}(x,A,B)$, the integral $I_{m}(x,A,B)$ and
$\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,x,A,B)$ for the special case of $m=1$ and
fixed values of parameters $A$ and $B$ (for RE Scarf-I: $A=4$, $B=2$ and for
RE GPT: $A=2$, $B=5$) are shown in Table:$2$. The one parameter family of
potentials $\hat{\phi}_{1,ext}(\lambda,x,A,B)$ and the normalized ground state
eigenfunctions $\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,x,A,B)$ corresponding to
these two potentials are given in Table $3$. The expressions for the
corresponding RE Pursey and the RE AM potentials for the same sets of
potential parameters are also given in Table $3$. The one parameter
($\lambda$) family of rationally extended scalar potentials for different
$\lambda$ and their corresponding ground state eigenfunctions for RE Scarf-I
and RE generalized Pöschl-Teller type scalar Potential are shown in Fig. $2$
and Fig. $3$ respectively.

Fig.2: (a) Plots of $\hat{\phi}_{1,ext}(\lambda,x,4,2)$ vs. $x$ for positive
$\lambda$ $(0,0.001,0.1,1$ and $\infty)$. The REP scalar potential is shown
for $\lambda=0$.

Fig.2: (b) Plots of $\hat{\phi}_{1,ext}(\lambda,x,4,2)$ vs. $r$ for negative
$\lambda(-\infty,-1.001,-1.01$ and $-1)$. The REAM scalar potential is shown
for $\lambda=-1$.

Fig.2: (c) Plots of normalized ground-state eigenfunctions
$\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,x,4,2)$ corresponding one parameter family
of scalar potentials (with positive $\lambda$)shown in Fig. $2(a)$.

$X_{m}$ Case |  RE Scarf-I |  RE GPT
---|---|---
$\phi_{con.}(x,A,B)$ |  $A\tan{x}-B\sec{x}$;
$0<B<A-1$, $-\frac{\pi}{2}<x<\frac{\pi}{2}$ |  $A\coth{r}-B{\rm cosech}{r}$;
$B>A+1>1$, $0<r<\infty$
$\phi_{m,rat.}(x,A,B)$ |  $-\frac{(\beta-\alpha+m-1)}{2}z^{{}^{\prime}}(x)\big{(}\frac{P^{(-\alpha-1,\beta+1)}_{m-1}(z(x))}{P^{(-\alpha-2,\beta)}_{m}(z(x))}\ \allowbreak-\frac{P^{(-\alpha,\beta)}_{m-1}(z(x))}{P^{(-\alpha-1,\beta-1)}_{m}(z(x))}\big{)}$; $z(x)=\sin{x},\break$
$\alpha=A-B-\frac{1}{2}$, $\beta=A+B-\frac{1}{2}$ |  $-\frac{(\beta-\alpha+m-1)}{2}z^{\prime}(r)\big{(}\frac{P^{(-\alpha-1,\beta+1)}_{m-1}(z(r))}{P^{(-\alpha-2,\beta)}_{m}(z(r))}-\frac{P^{(-\alpha,\beta)}_{m-1}(z(r))}{P^{(-\alpha-1,\beta-1)}_{m}(z(r))}\big{)};$ $z(r)=\cosh{r}$
$\alpha=-A+B-\frac{1}{2}$, $\beta=-A-B-\frac{1}{2}$
$\Psi^{(1)}_{n,m,ext}(x,A,B)$ |  $N^{(1)}_{n,m}\frac{(1-z(x))^{\frac{(A-B)}{2}}(1+z(x))^{\frac{(A+B)}{2}}}{P^{(-\alpha-1,\beta-1)}_{m}(z(x))}\times\ \allowbreak\hat{P}^{(\alpha,\beta)}_{n+m}(z(x))$ |  $N^{(1)}_{n,m}\frac{(z-1)^{\frac{(B-A)}{2}}(z+1)^{-\frac{(B+A)}{2}}}{P^{(-\alpha-1,\beta-1)}_{m}(z(r))}\times\ \allowbreak\hat{P}^{(\alpha,\beta)}_{n+m}(z(r))$
$E^{(1)}_{n}$ |  $(A+n)^{2}-A^{2};\quad n=0,1,2..$ |  $A^{2}-(A-n)^{2};\quad n=0,1,2..$

Table $1$: Expressions of Extended Lorentz scalar potentials
$(\phi_{m,ext}(x,A,B)=\phi_{con.}(x,A,B)+\phi_{m,rat.}(x,A,B))$, first
component of the eigenfunctions $(\Psi^{(1)}_{n,m,ext}(x,A,B))$ and their
energy eigenvalues $(E_{n}^{(1)})$ corresponding to RE Scarf-I and RE GPT like
potentials. Here $\hat{P}_{n+m}^{(\alpha,\beta)}(z)$ is the
$X_{m}$-exceptional Jacobi orthogonal polynomial.

$X_{1}Case$ |  RE Scarf-I |  RE GPT
---|---|---
$\phi_{1,ext}(x,A,B)$ |  $\Bigl{(}4\tan(x)-2\sec(x)\newline +\frac{8\cos(x)}{-71+8\cos(2x)+64\sin(x)}\Bigr{)}$ |  $\Bigl{(}2\coth{r}-5{\rm cosech}{r}+10\sinh r\times\allowbreak\big{(}\frac{1}{10\cosh r-5}-\frac{1}{10\cosh r-3}\big{)}\Bigr{)}$
$\Psi^{(1)}_{0,1,ext}(x,A,B)$ |  $(\frac{8}{3}\sqrt{\frac{10}{39\pi}})\frac{(1-z(x))(1+z(x))^{3}}{P^{(-\frac{5}{2},\frac{9}{2})}_{1}(z(x))}\hat{P}^{(\frac{3}{2},\frac{11}{2})}_{0+1}(z(x))$ |  $(21\sqrt{\frac{11}{2}})\frac{(z-1)^{\frac{(3)}{2}}(z+1)^{-\frac{(7)}{2}}}{P^{(-\frac{7}{2},-\frac{17}{2})}_{1}(z(r))}\hat{P}^{(\frac{5}{2},-\frac{15}{2})}_{0+1}(z(r))$
$I_{1}(x,A,B)$ |  $\Big{(}\frac{1}{32760\pi(-7+4\sin(x))}(-114660(\pi+2x)+244608\cos(x)+59696\cos(3x)$
$+11984\cos(5x)-854\cos(7x)$
$-42\cos(9x)+65520(\pi+2x)\sin(x)$
$-125216\sin(2x)+3416\sin(4x)$
$+5984\sin(6x)+141\sin(8x))\Big{)}$ |  $\big{(}85+1103\cosh(r)+178\cosh(2r)+19\cosh(3r)+\cosh(4r)\big{)}\times\frac{{\rm sech}^{6}(\frac{r}{2})\tanh^{7}(\frac{r}{2})}{(-32+64\cosh(r))}$

Table 2: The $X_{1}$ cases $(m=1)$ for RE Scarf-I and RE GPT scalar
potentials, their ground state eigenfunctions $\Psi^{(1)}_{0,1,ext}(x,A,B)$
and integral $I_{1}(x,A,B)$ for fixed values of the parameters $(A=4,B=2)$ and
$(A=2,B=5)$ respectively.

$X_{1}Case$ |  RE Scarf-I |  RE GPT
---|---|---
$\hat{\phi}_{1,ext}(\lambda,x,A,B)$ |  $-\frac{2(G(x)+M(x)+16380(137+2S(x))(\pi+2x+2\pi\lambda))\sec(x)}{(-9+4\sin(x))(H(x)-114660(\pi+2x+2\pi\lambda)+1456D(x)\cos(x)+131040\pi\lambda\sin(x))}$
|  $2\coth(r)-5{\rm cosech}(r)+\frac{4\sinh(r)}{3+4\cosh(r)(-4+5\cosh(r))}$
$+\frac{198(3-10\cosh(r))^{2}{\rm
cosech}(r)\sinh^{7}\left(\frac{r}{2}\right)}{(-1+2\cosh(r))\left(32\lambda\cosh^{13}\left(\frac{r}{2}\right)(-1+2\cosh(r))+Q(r)\sinh^{7}\left(\frac{r}{2}\right)\right)}$;
$Q(r)=85+1103\cosh(r)+178\cosh(2r)+19\cosh(3r)+\cosh(4r)$
$\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,x,A,B)$ |  $\frac{\sqrt{\lambda(1+\lambda)}}{\big{(}I_{1}(x,4,2)+\lambda\big{)}}(\frac{8}{3}\sqrt{\frac{10}{39\pi}})\frac{(1-z(x))(1+z(x))^{3}}{P^{(-\frac{5}{2},\frac{9}{2})}_{1}(z(x))}\hat{P}^{(\frac{3}{2},\frac{11}{2})}_{0+1}(z(x))$;
with $z(x)=\sin{x}$ |  $\frac{\sqrt{\lambda(1+\lambda)}}{\big{(}I_{1}(x,2,5)+\lambda\big{)}}(21\sqrt{\frac{11}{2}})\frac{(z-1)^{\frac{(3)}{2}}(z+1)^{-\frac{(7)}{2}}}{P^{(-\frac{7}{2},-\frac{17}{2})}_{1}(z(r))}\hat{P}^{(\frac{5}{2},-\frac{15}{2})}_{0+1}(z(r))$;
with $z(r)=\cosh{r}$
$\phi^{[P]}_{1,ext}(x,A,B)$ |  $-\frac{2(G(x)+M(x)+16380(137+2S(x))(\pi+2x))\sec(x)}{(H(x)-114660(\pi+2x)+1456D(x)\cos(x))(-9+4\sin(x))}$ |  $2\coth\left(\frac{r}{2}\right)+\frac{10\sinh(r)}{3-10\cosh(r)}-3\tanh\left(\frac{r}{2}\right)$
$+\frac{1103\sinh(r)+356\sinh(2r)+57\sinh(3r)+4\sinh(4r)}{85+1103\cosh(r)+178\cosh(2r)+19\cosh(3r)+\cosh(4r)}$
$\phi^{[AM]}_{1,ext}(x,A,B)$ |  $-\frac{2(G(x)+M(x)-16380(137+2S(x))(\pi-2x))\sec(x)}{(-9+4\sin(x))(H(x)+114660(\pi-2x)+1456D(x)\cos(x)-131040\pi\sin(x))}$;
with
$M(x)=-5000996\cos(x)+780528\cos(3x)+50540\cos(5x)+29003\cos(7x)+2345\cos(9x)+84\cos(11x)+3122840\sin(2x)$,
$S(x)=-35\cos(2x)-107\sin(x)+4\sin(3x)$,
$G(x)=-208840\sin(4x)-49335\sin(6x)+685\sin(8x)+177\sin(10x)$,
$H(x)=59696\cos(3x)+11984\cos(5x)-854\cos(7x)-42\cos(9x)+3416\sin(4x)+5984\sin(6x)+141\sin(8x)$, $D(x)=168-172\sin(x)+45(\pi+2x)\tan(x)$ |  $2\coth(r)-5{\rm cosech}(r)+\frac{4\sinh(r)}{3+4\cosh(r)(-4+5\cosh(r))}$
$+\frac{198(3-10\cosh(r))^{2}{\rm
cosech}(r)\sinh^{7}\left(\frac{r}{2}\right)}{(-1+2\cosh(r))\left(-32\cosh^{13}\left(\frac{r}{2}\right)(-1+2\cosh(r))+Q(r)\sinh^{7}\left(\frac{r}{2}\right)\right)}$;
$Q(r)=85+1103\cosh(r)+178\cosh(2r)+19\cosh(3r)+\cosh(4r)$

Table $3$: The expressions for $\hat{\phi}_{1,ext}(\lambda,x,A,B)$, the RE
Pursey and the RE AM scalar potentials and their corresponding ground state
eigenfunctions ($\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,x,A,B)$) for RE Scarf-I
and RE GPT scalar potential for fixed values of the parameters $(A=4,B=2)$ and
$(A=2,B=5)$ respectively.

Fig.3: (a) Plots of $\hat{\phi}_{1,ext}(\lambda,r,2,5)$ vs. $r$ for positive
$\lambda(0,0.001,0.01,1$ and $\infty)$. The REP potential is shown for
$\lambda=0$.

Fig.3: (b) Plots of $\hat{\phi}_{1,ext}(\lambda,r,2,5)$ vs. $r$ for negative
$\lambda(-\infty,-1.1,-1.01,-1.001$ and $-1)$. The REAM scalar potential is
shown for $\lambda=-1$.

Fig.3: (c) Normalized ground-state eigenfunctions
$\hat{\Psi}^{(1)}_{0,1,ext}(\lambda,r,2,5)$ for some potentials (with positive
$\lambda$)shown in Fig. $3(a)$.

##  4 Summary and Conclusions

In this work, we have considered the one-dimensional Dirac equation with three
different RE Lorentz scalar potentials, i.e. Radial oscillator, the Scarf-I
and GPT scalar potentials and for all of them have constructed a one
continuous parameter family $(\lambda)$ of strictly isospectral RE Lorentz
scalar potentials and obtained their solutions in terms of $X_{m}$-exceptional
orthogonal polynomials. The cases of RE Pursey and the RE AM are also
discussed in the special cases of $\lambda=0$ and $-1$ respectively. The
ground and the excited state solutions of all these RE strictly isospectral
scalar potentials are obtained explicitly. For the special case of $m=1$, we
have graphically shown the behavior of the strictly isospectral potentials as
a function of the continuous parameter $\lambda$.

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