# The evolution of pluralistic ignorance

Alessandra F. Lütz<EMAIL_ADDRESS>Departamento de Física, Universidade
de Minas Gerais, 31270-901, Belo Horizonte MG, Brazil  Lucas Wardil
Departamento de Física, Universidade de Minas Gerais, 31270-901, Belo
Horizonte MG, Brazil

(August 28, 2024)

###### Abstract

Pluralistic ignorance is a social-psychological phenomenon characterized by
individuals privately maintaining beliefs or opinions that diverge from what
they perceive to be the prevailing norm within a group or society. Typical
models are based on opinion dynamics involving both private and public states,
where agents’ binary states undergo changes influenced by social pressure.
However, these models overlook a crucial aspect of pluralistic ignorance: if
the absence of behavioral expression matches the normative status of the
behavior, social pressure exerted by the initial group configuration is enough
to induce pluralistic ignorance. Here, we show that pluralistic ignorance is
maintained if imitation of others is not too frequent and the social influence
of the initial minority is high. Interestingly, the individuals are able to
overcome the pluralistic ignorance by the end of interactions, yet it
resurfaces at the outset of each subsequent group interaction. However, if the
likelihood of individuals imitating others becomes excessively high,
pluralistic ignorance completely dissipates in an undesirable manner:
individuals internalize the dysfunctional behavior. We also show that, if
memory is added to the internalization process, pluralistic ignorance reaches
high levels only for intermediate imitation rates.

††preprint: APS/123-QED

##  I Introduction

Pluralistic ignorance refers to a social phenomenon where individuals
privately hold beliefs or opinions that differ from the perceived norm within
a group or society and assume, incorrectly, that their beliefs differ from
everyone else’s [1]. In other words, it is a situation where people conform to
what they believe others think or feel, even though those beliefs might not
align with their personal states. This phenomenon often occurs because
individuals are concerned about social acceptance and don’t want to be seen as
deviating from the perceived norm [2, 3, 4, 5]. As a result, people may keep
their true beliefs or feelings hidden, assuming that they are in the minority
or that their opinions are unpopular. This can lead to a collective
misunderstanding of what the group actually believes, as everyone is privately
concealing their true thoughts. Pluralistic ignorance can have important
implications for group dynamics, decision-making, and social behavior, as it
can lead to the perpetuation of norms and beliefs that might not actually
reflect the majority’s true sentiments. It can also hinder open and honest
communication within a group or society [6, 7].

In the social-psychological literature, two classic examples are typically
cited to illustrate this phenomenon. The first is the classroom scenario [8,
9]. After a teacher has concluded a presentation of challenging content in
class and invites questions from the students, each student refrains from
asking a question despite not having a complete grasp of the material. This
silence stems from the observation that none of the students in the class have
asked a question. Consequently, each student assumes that everyone else
believes the material was not difficult, except for themselves. In an effort
to avoid the prospect of publicly appearing as the sole individual who did not
comprehend the material, no student is willing to pose a question. The second
example often cited is the college drinking scenario [4, 10]. A group of
first-year students has recently moved into their new dormitory. The
prevailing campus culture promotes the notion that excessive drinking is
socially desirable. During the first party, each student partakes in heavy
drinking, despite personally finding it unpleasant.

These two examples have some important differences. In the classroom example,
the initial silence is shared by those who are willing to ask a question but
have not expressed it themselves yet and also by those who are afraid of
exposing their questions. The initial state of the group is characterized by
inaction and is interpreted as a social norm. In contrast, in the drinking
example, the individuals are expected to conform to university campus drinking
behavior and, despite the obvious fact that at the beginning of a party, the
drinking has to be started, they readily engage in excessive drinking
behavior. Thus, the initial inaction cannot be interpreted as a social norm of
not drinking. This difference relies upon the difference between injunctive
and descriptive norms [11]. Observing common actions and perceiving widespread
approval from others serve as distinct sources of motivation for humans.
Injunctive norms are about how the group expects one member to behave, whereas
descriptive norms are about what is observed as the prevalent behavior in the
group. The classroom situation is better described as a descriptive norm
instance because students observe others not asking and infer from this that
they will not expose themselves as unfit. On the other hand, the drinking
situation is an injunctive instance because there is a normative status saying
that drinking is cool, making part of the liberal culture of campuses [4]. In
our work, we explore this difference. But let us first review how pluralistic
ignorance has been modeled in the specialized literature.

Although Pluralistic Ignorance has been a subject of scientific inquiry since
1924 [12], with the term originating in 1931 [13], there is still a notable
absence of specialized mathematical models dedicated to exploring this
phenomenon. The existing models draw upon opinion modeling frameworks, which
have attracted substantial attention across several scientific disciplines
such as sociology, physics, and political science [14, 15]. Various types of
opinion models have emerged, from the Sznajd opinion models [16], which
investigate peer influence and consensus formation, to Axelrod’s cultural
dissemination model [17], which explores the diffusion of cultural traits.
Additionally, the classic voter model [18], in which individuals imitate the
most popular opinion among their immediate neighbors, is widely used as a
framework for studying opinion dynamics and social influence.

Many interesting variations of the classic voter model have been developed
[19, 20, 21, 22], ranging from extensions that incorporate more than two
strategies [23, 24, 25, 26, 27] to versions that incorporate unconventional
behaviors, like the voter model with zealots [28, 22, 29, 30], where a subset
of individuals adhere steadfastly to a particular opinion, or a voter model
with contrarians [31, 32], which are individuals who actively oppose the
majority sentiment. The voter model has also been adapted to account for the
impact of external influences, resulting in models with noise [32], capturing
the stochastic nature of real-world opinion dynamics. Furthermore,
advancements include the integration of memory [33], allowing individuals to
consider past interactions when updating their opinions. Additionally,
different types of network structures [34, 19, 35, 36], such as scale-free
[37, 38, 39, 40] or small-world networks [41, 42], have been incorporated to
study how the underlying connectivity influences the spread of opinions,
consensus formation, and critical behavior.

Because pluralistic ignorance involves a dissonance between people’s public
actions and their true beliefs or feelings, a pluralistic ignorance model
should consider hidden opinions. One opinion model that considers public and
private opinions is the partisan voter model [43, 44], in which agents have
innate fixed preferences that influence their expressed opinions. A different
approach, where the hidden opinions may be affected by the expressed ones, is
the voter model with concealed opinions [21]. An extension of this model is
used in [45] to study hypocrites, that is, to study individuals whose internal
and external opinions differ. Notice that, although hypocrisy is a necessary
condition for pluralistic ignorance, it is not a sufficient one. This is so
because hypocrisy may be caused by different psychological phenomenons, not
all being related to pluralistic ignorance [46].

Pluralistic ignorance was explicitly modeled using an agent-based model in
[47, 48, 49], where the authors propose an adaptation of the Voter [18] and
the Friedkin-Johnsen [50] models. The proposed model adopts a two-layered
network framework, distinguishing between internal opinions (beliefs) and
external/public opinions. This design aims to investigate the manifestation of
pluralistic ignorance across various scenarios and network topologies. The
analysis focuses on two primary approaches. One is based on peer interactions
[48, 47], where agents imitate each other. The other approach focuses on the
group perspective and uses payoffs to balance the need to conform to the group
identity and the need to minimize cognitive dissonance [49, 47]. In both
approaches public opinions can be internalized if agents are susceptible
enough to external influences and the pressure to conform is low enough.
Pluralistic ignorance arises when the pressure to conform is above a
threshold, forcing agents to adopt public opinions that are different from
their internal beliefs.

In Seem’s model, however, like in the traditional opinion models, the
individuals display their external state continuously: the external state of
others are always available for scrutiny. Here, we depart from a simpler
situation, which is better represented by the classroom example. Not revealing
the true internal state is equivalent to adopting the inaction external state.
For example, although someone may want to ask a question, not revealing it
means, for others, that the true state is “not having any question at all”.
Thus, at the beginning of the interaction, everyone effectively adopts the no-
action state as the external state and this unanimity sends the message that
this is the group norm. However, as the individuals externalize their internal
action state, the others may decide to follow the group and the decision of a
few may influence the decision of others in the sequential process. As time
passes, the individuals may internalize their external state, changing the
composition of the population. But most importantly, every time a group of
strangers is formed the same initial configuration of inaction takes place
and, depending on the internal state of the individuals, pluralistic ignorance
can emerge. There is a second difference from Seem’s approach. In [47, 48, 49]
the proxy for pluralistic ignorance is the fraction of individuals whose
internal state differs from the external state. Although we agree that this a
necessary condition, some reference to a group norm must be present in the
measurement. In our study, we adopt a different approach to quantify
pluralistic ignorance.

Here, we model the pluralistic ignorance phenomenon based on descriptive norms
by setting a pair of internal and external states for each individual. Because
the initial inaction of all individuals in the group can be interpreted as the
norm, those that have an internal state corresponding to an action state (for
example, willingness to ask a question), are said to be under pluralistic
ignorance. The individuals can imitate the prevailing observed behavior in the
group or can externalize their internal state. At the end of the interaction,
they can internalize the external state. We show that pluralistic ignorance
emerges and ceases only if the imitation probability is high. However, at the
same time, many individuals are able to overcome pluralistic ignorance, with a
few remaining with an internal state incompatible with the external one. We
also investigate the effect of memory in the internalization process by
allowing the individuals to internalize the most frequent external state
adopted in the last interactions.

The paper is structured as follows: We begin by defining the model in the
subsequent section. The results section is divided into three subsections.
Firstl, we examine the scenario where groups are randomly selected from a
population of individuals with no memory. Next, we incorporate memory into the
model. Third, we place individuals as nodes on a square lattice to explore the
impact of correlations. Finally, we conclude by discussing the key concepts,
their validity, and potential further applications.

##  II Model

We consider a population of size $N$ where each individual has an internal
state $s_{i}$ and an external state $s_{e}$. The states can be either $0$ or
$1$. Thus, each individual is characterized at a given time by the vector
$(s_{i},s_{e})$. For example, individuals of type $(0,1)$ or $(1,0)$
externalize a state different from their internal one. The external state
represents the displayed behavior. For example, the external state $1$ can
represent the action of asking a question and the $0$ the inaction. The
internal state represents the attitude, belief, or inclination toward the
behavior. For example, the internal state $1$ represents the inclination to
ask a question and the $0$ represents indifference or a lack of inclination to
inquire.

The dynamics involve a stochastic process where a group of $G$ agents is
randomly selected, and each agent is given the opportunity to update the
external state. Initially, all individuals in the group have their external
state set to $0$. The sequence in which individuals are considered for update
is completely randomized. So, a random individual is chosen to imitate the
external state of the majority in the group with probability $p_{c}$ or
externalizes his internal state with probability $1-p_{c}$. If the imitation
takes place, then with probability $p_{1}$, given by

$p_{1}=\left(\frac{n_{1}}{G}\right)^{1-\gamma},$

the external state of the individual doing the update is set to $1$. Here,
$n_{1}$ is the number of individuals in the group with an external state equal
to $1$ and $\gamma$ is a parameter controlling the influence of the initial
minority ($0<\gamma\leq 1$). If $\gamma$ is high, then a single individual
externalizing $1$ can trigger the adoption of the state $1$ by others. Notice
that $n_{1}$ starts equal to zero and can increase as more individuals
externalize the state $1$. However, the first appearance of the external state
$1$ in the group is only possible if an individual with an internal state
equal to $1$ externalizes it. We should emphasize that the initial state of
$s_{e}=0$ for everyone at the onset of group interaction signifies universal
inaction among all participants, which represents the descriptive norm working
in the group.

Individuals can internalize their external state. Each individual has a memory
of size $T_{M}$ where they store the last $T_{M}$ external states. At the end
of the group interactions, each individual internalizes the most common state
in his memory with probability $p_{int}$. For example, if $T_{M}=1$ the
external state adopted in the current round is the one that can be immediately
internalized. We say that the $T_{M}=1$ case is the memoryless case. Figure 1
shows the possible transitions in the individual state space
$\\{(0,0),(0,1),(1,0),(1,1)\\}$.

The population evolution is simulated as follows. First, we initialize half of
the individuals with internal state equal to $1$, and the remaining half to
$0$. Then, we repeat the group update process $N/G$ times so that each
individual has a chance to participate on average at least in one group. Every
time an individual is chosen to be part of a group, his external state is set
to zero. These $N/G$ repetitions comprise a Monte Carlo Step (MCS) of the
simulation. At the end of each MCS, we measure the fraction of each type of
external and internal state, as well as the individual configuration, given by
$(s_{i},s_{e})$. Notice that the external state is given by the value it
assumed in the most recent participation in a group and that this value is set
to zero every time a new group is formed. In all figures presented in the
paper, we show stationary values obtained by the time averaging after a
transient period.

Figure 1: Diagram of the state transitions for memoryless agents ($T=1$). The
state of one individual is represented by the vector $(s_{i},s_{e})$, where
$s_{i}$ is the internal state and $s_{e}$ is the external state. During the
round, individuals in the $(0,0)$ state can transition to $(0,1)$ by imitating
the state $1$ and to $(1,1)$ by internalizing the imitated state. Similarly,
$(1,0)$ individuals can transition to $(1,1)$ by imitating state 1 or by
externalizing the state 1. The $(1,0)$ individuals can also transition to
$(0,0)$ by internalization. The shaded states on the left represent the
initial states at the beginning of group interactions.

The pluralistic ignorance level is measured as the fraction of individuals
with $s_{i}=1$. Because at the beginning of the group interaction, everyone
displays $s_{e}=0$, due to social pressure, an individual with $s_{i}=1$ will
have the impression that they may be the only one with internal state
$s_{i}=1$. At the same time, after all the interactions in the group take
place, the fraction of individuals with $s_{i}=1$ and $s_{e}=1$ represents
those that manage to overcome the pluralistic ignorance, expressing their
internal state despite the initial opposition. The individuals that, after the
group interactions, end with $s_{i}=0$ and $s_{e}=1$, or conversely, with
$s_{i}=1$ and $s_{e}=0$, can be interpreted as individuals experiencing
conflict between internal and external states caused by the tentative to
conform to the behavior of the majority. The individuals that end up
internalizing the $0$ external state, starting the group interactions in the
state $(0,0)$, do not feel any psychological discomfort or pluralistic
ignorance. However, if we interpret the inaction as a dysfunctional outcome,
the $(0,0)$ is an unwanted outcome.

##  III Results

We considered a system with $N=484$ agents and groups of size $G=20$. For the
initial configuration of the population, the internal states were randomly
initialized to either $0$ or $1$ with equal probability. Recall that the
external states are set to $0$ at the beginning of every group interaction.

Figure 2 shows the level of pluralistic ignorance for various combinations of
imitation probability ($p_{cp}$) and influence factor ($\gamma$). Pluralistic
ignorance occurs only when $\gamma$ is high enough and $p_{cp}$ is not too
high. In the absence of imitation ($p_{cp}=0$), the internal states do not
change. Alterations occur only after an individual internalizes an imitated
state, and this cannot happen for $p_{cp}=0$. Thus, the fraction of
individuals with an internal state equal to $1$ and $0$ remains equal to the
initial fractions (both of which are set at $0.5$). If $p_{cp}$ is small, most
individuals externalize their internal state. This means that it is very
likely that an individual in the state $(1,0)$ changes to the state $(1,1)$.
From this point on, if the influence factor $\gamma$ is sufficiently high, it
is the state 1 that is going to be imitated. Therefore, individuals in the
state $(0,0)$ imitate state $1$ and eventually internalize it, becoming
$(1,1)$, which increases the fraction of individuals with an internal state
equal to $1$. If the imitation probability is too high, there are limited
opportunities for someone to externalize the state $1$ (recall that with
probability $1-p_{cp}$ the individual externalizes his internal state). As a
result, the majority of individuals tend to imitate the state $0$, eventually
internalizing it. This leads to a final state of homogeneity, where all
individuals possess an internal state equal to 0. In this case, there is no
chance for any $1$ to appear and, although there is no pluralistic ignorance,
the population is sadly trapped in the state $(0,0)$.

For the external state $1$ to propagate, not only does the imitation
probability have to be lower than a threshold value, but also the influence
factor $\gamma$ has to be significant. After the first individual externalizes
the state $1$, the probability of the subsequent individual imitating this
behavior is equal to ${(1/G)^{1-\gamma}}$. This probability increases with
$\gamma$. Thus, a high value of the influence factor is required to provide a
minimal chance for the external state $1$ to be imitated. If $\gamma$ is low,
the population is absorbed in the configuration where everyone in the state
$(0,0)$.

Figure 2: Pluralistic ignorance level. The top figure shows the fraction of
pluralistic ignorance as a function of the imitation probability ($p_{cp}$)
and the influence factor ($\gamma$). The bottom figure shows a cross-section
of the top figure at $\gamma=0.99$.

To understand the role played by the first individual that externalizes the
state $1$, we can use a mean-field approach to calculate the mean number of
individuals displaying $1$ as their external state. First, let us define two
auxiliary functions:

$\displaystyle P(k)$ $\displaystyle=$ $\displaystyle
p_{cp}\left(\frac{k}{G}\right)^{1-\gamma}+(1-p_{cp})x_{1}$ (1) $\displaystyle
Q(k)$ $\displaystyle=$ $\displaystyle
p_{cp}\left(1-\left(\frac{k}{G}\right)^{1-\gamma}\right)+(1-p_{cp})x_{0}$ (2)

where $x_{0}$ and $x_{1}$ are the fraction of individuals with internal state
equal to 0 and 1 in the population, respectively. Thus, $P(k)$ is the
probability that a new external state equal to $1$ appears, whereas $Q(k)$ is
the probability that an external state equal to $0$ is maintained. In the
first round, the first individual externalizes the state $1$ with probability
$P(0)$ and the state $0$ with probability $Q(0)$. In the second round, there
are four possibilities for the choices of the first and second individuals:
$11$, $10$, $01$, and $00$, where the first digit represents the state of the
first individual to display their external state. Each one of these
possibilities happens with probability given by $P(11)=P(0)P(1)$,
$P(10)=P(0)Q(1)$, $P(01)=Q(0)P(0)$, and $P(00)=Q(0)Q(0)$, respectively. Thus,
if there were only two individuals in the group, the average number of
individuals displaying an external state equal to $1$ is given by

$\bar{q}_{2}=2\cdot P(11)+1\cdot[P(10)+P(01)].$ (3)

Let us generalize this expression. If $s_{1}$ is the state of the first
individual in the group, then the probability of the outcome $s_{1}$ is given
by

$P_{1}(s_{1})=s_{1}P(0)+(1-s_{1})Q(0).$ (4)

The probability of the outcome $s_{1}s_{2}$, where $s_{2}$ is the external
state displayed by the second individual in the random order in the group, is
given by

$P_{2}(s_{1},s_{2})=P_{1}(s_{1})[s_{2}P(s_{1})+(1-s_{2})Q(s_{1})].$ (5)

Thus,

$\displaystyle P_{t}(s_{1},\ldots,s_{t})$ $\displaystyle=$ $\displaystyle
P_{t-1}(s_{1},\ldots,s_{t-1})\left[s_{t}P\left(\sum_{i-1}^{t-1}s_{i}\right)\right.$
$\displaystyle\left.+(1-s_{t})Q\left(\sum_{i-1}^{t-1}s_{i}\right)\right]$

and

$\bar{q}_{t}=\sum_{s_{1},\ldots,s_{t}}\left(\sum_{i=1}^{t}s_{i}\right)P_{t}(s_{1},\ldots,s_{t}).$
(6)

Figure 3 shows the mean number of individuals displaying 1 for $x_{1}=0.2$. As
we raise the value of $p_{cp}$, the average number of individuals displaying
the external state $1$ initially increases, reaching a peak, and then starts
to decrease, falling below a count of one. This peak only occurs when $\gamma$
is sufficiently high, underscoring the necessity for stimulating imitation to
sustain the external presence of the state $1$. Furthermore, the average count
drops below one when $p_{cp}$ approaches $0.8$, which represents the threshold
at which internal state 0 prevails within the population

Figure 3: Mean number of individuals displaying external state equal to $1$
for a fraction of individuals with internal opinion $1$ equal to $0.2$. From
top to bottom: $\gamma=0.99,0.9,0.5,0.0$. The horizontal line is the baseline.

The individuals overcome pluralistic ignorance if those having an internal
state equal to $1$ are able to externalize their internal state, breaking
through the barrier of social pressure. To see this, we measure the fraction
of each configuration $(s_{i},s_{e})$ after all individuals have the
opportunity to participate in a group and externalize, imitate, or
internalize. Figure 4 shows the fraction of individuals in each state after
the populations reaches the stationary state. If $p_{cp}$ is not too high,
many $(1,0)$ individuals manage to change to $(1,1)$. However, although these
$(1,1)$ individuals temporarily overcome the pluralistic ignorance, they will
keep experiencing it because they start in the state $(1,0)$ in the next group
interaction. As $p_{cp}$ increases, the fraction of individuals that manage to
get to the $(1,1)$ state decreases until all individuals become in the
harmonized state $(0,0)$. Interestingly, the fraction of individuals
experiencing conflict between internal and external states at the end of the
group interaction increases with $p_{cp}$, reaches a peak near $0.8$, and
drops to zero afterward. The reason of this non-monotonic behavior is simple.
For low $p_{cp}$ the externalization is very frequent and most individuals end
the interactions harmonized, either as $(0,0)$ or as $(1,1)$. As $p_{cp}$
increases, there is more imitation and individuals $(0,1)$ and $(1,0)$ emerge.
If $p_{cp}$ is too high, the external state equal to $1$ is not displayed by
anyone and, due to the internalization, everyone eventually internalize the
external state $0$. Therefore, there must be a maximum at some intermediate
value of $p_{cp}$, which happens to be close to $0.8$.

Figure 4: Fractions of each type $(s_{i},s_{e})$ at the end of the group
interactions. Parameters are $p_{int}=0.01$ and $\gamma=0.9$.

The internalization probability is crucial to allow the modification of the
internal states. Our results are qualitatively unaffected by $p_{int}$. As
$p_{int}$ increases, the fraction of dissonant states (states $(0,1)$ and
$(1,0)$) at the end of group interactions is reduced because individuals
internalize the external state, becoming harmonized more often. This mechanism
can slightly change the fractions of $(1,0)$ and $(0,1)$ during the transient
regime, but do not affect the results qualitatively. The most pronounced
effect being the decrease in the $p_{cp}$ above which the $(0,0)$ state
dominates the population.

As a final observation, one might consider why individuals, having overcome
pluralistic ignorance (as evidenced by many ending up at $(1,1)$ for $p_{cp}$
not too high), don’t retain that information and immediately begin the next
interaction displaying $s_{e}=1$. This could be the case if the external state
were merely a visible, permanent feature as in traditional opinion models.
However, the root of pluralistic ignorance in our model lies precisely in the
initial absence of behavior expression by everyone at the start of
interactions, interpreted as a descriptive norm enforced by social pressure.

###  III.1 Internalization with memory

How intense must an external state be experienced to enable its
internalization, thereby transforming it into a permanent aspect of an
individual’s being? It’s worth considering that individuals might not
internalize the current external state, but rather the most common state
observed. To investigate this point, we added memory to the model. Each
individual stores the last $T_{M}$ external states adopted by them and, with
probability $p_{int}$, they internalize the most frequent one.

Figure 5 shows the level of pluralistic ignorance for different memory
lengths. For all memory lengths, the level of pluralistic ignorance increases
as $p_{cp}$ increases, reaches a plateau of high value, and goes to zero at a
high $p_{cp}$ value, mirroring the behavior observed previously for $T_{M}=1$.
The main difference from the $T_{M}=1$ case is that the pluralistic ignorance
is high only at intermediate values of $p_{cp}$. The main reason for this
behavior is that for $T_{M}>1$ the individuals can internalize a state that is
different from the state displayed in the last round. More specifically, the
$(0,0)$ individuals that become $(0,1)$ must internalize the state 1 to become
$(1,1)$. However, because the probability of imitation is low, the $(0,0)$
individuals will remain as $(0,0)$ most of the time, and the memory will be
full of $0$’s.

Figure 5: The top figure shows the pluralistic ignorance level as a function
of the imitation probability $p_{cp}$ for different memory lengths. The bottom
figure shows the frequencies of $(s_{i},s_{e})$ at the end of the group
interactions as functions of the imitation probability $p_{cp}$ for a memory
length of $T_{M}=10$. Here the internalization probability is $p_{int}=0.01$
and the influence factor is $\gamma=0.9$.

The effect of the memory is more intricate. For $T=1$, the fraction of $(1,0)$
and $(0,1)$ at the end of the group interactions are equal. However, as shown
in Fig. 5, for $T_{M}>1$ these fractions are no longer the same. Figure 6
shows in detail this asymmetry introduced by memory. As can be seen in the
first panel of Fig. 6, the difference in the frequencies of states $(0,1)$ and
$(1,0)$ becomes greater as the memory length, $T_{M}$, increases. There are
more $(0,1)$ states for low values of $p_{cp}$, the opposite happening for
high $p_{cp}$. Not only does this difference increase with $T_{M}$, but also,
the value of $p_{cp}$ above which $(1,0)$ becomes higher than $(0,1)$
increases. The main reason for such behavior is indicated in the second panel
of Fig. 6, where the frequency of internalization events in which the
internalized state is different from the last external state is shown.

Figure 6: Dissonance. The left panel shows the difference between the
frequencies of dissonant states $(0,1)$ and $(1,0)$ as a function of the
imitation probability $p_{cp}$ for different memory lengths. The right panel
shows the frequency of internalization events that involve internalizing a
state different from the current external one. The inset in the right panel
shows the levels of dissonant states for the same system. Here, $p_{int}=0.01$
and $\gamma=0.9$.

###  III.2 Square lattice

Finally, we tested the effect of adding spatial correlations to the model. We
set each individual to a different site of a square lattice. At each time
step, one individual is randomly chosen to be the center of a group of size
$G=21$, where the interaction takes place. The pluralistic ignorance level is
very similar to the well-mixed case, shown in Fig. 2. This is to be expected
since the external states are set to zero at the beginning of each group
interaction. Thus, there are not many opportunities for the formation of
external state clusters, as in opinion dynamic models.

Figure 7 shows the spatial correlations of the internal state of the
individuals in the square lattice. The spatial correlations are nearly the
same for different distances, $R$. However, as the imitation probability
$p_{cp}$ changes, the mean correlation varies. The inset in the first panel
shows that the spatial correlation for $R=8$ has two maximum values: one for a
small value of $p_{cp}$, and another for a high $p_{cp}$, just before the
correlation reaches zero. For $p_{cp}=0.2$, external state $1$ is more
abundant than 0. As $p_{cp}$ increases, the difference in the amount of
external states $1$ and $0$ decreases, until state $0$ starts to be more
popular than 1. These two extremes, one with state $0$ being way more
frequent, and the other where $1$ is the most popular state, explain the
maximum correlation values observed in the first panel.

Figure 7: Spatial correlation and configurations for the internal opinions in
the square lattice. At the left, the multiple lines show the behavior of the
spatial correlation as a function of the distance for different values of the
imitation probability $p_{cp}$. The inset shows the spatial correlation as a
function of $p_{cp}$ for $R=8$. Notice that, since the system is not that much
larger than the group size, there is almost no variation in the spatial
correlation as the distance changes. At the right, four different spatial
configurations are shown, each one related to a different $p_{cp}$ value.
Here, $\gamma=0.9$, $p_{int=0.01}$, $T_{M}=1$ and the number of neighbors for
each agent is 20.

##  IV Discussion and Conclusions

In this work we introduced a simple model, similar to external and concealed
opinion models, to investigate the emergence of pluralistic ignorance. Our
results show that, in the absence of memory, pluralistic ignorance occurs for
a wide range of values of imitation probability if the influence factor is
strong. However, if there is memory, pluralistic ignorance emerges at high
levels only for moderate values of imitation probability. Interestingly, if
the imitation probability is not too high, at the end of the group
interactions the population reaches a stationary state where most individuals
are able to overcome the pluralistic ignorance. However, the individuals
return to pluralistic ignorance at the beginning of the next group
interaction. Most interestingly, the fraction of individuals that end the
group interaction phase with conflicting internal and external states reaches
a peak at an intermediate value of the imitation probability. If imitation
probability is too high, the individuals eventually internalize the inaction
and harmonize themselves in the state $(0,0)$.

We show that pluralistic ignorance can happen every time a group of strangers
meet. This is why we used the fraction of individuals with $s_{i}=1$ before
the interactions take place as a proxy for pluralistic ignorance in the
population (recall that $s_{e}=0$ is always the initial external state of
everyone in the group). The fact that many in the group can overcome the
pluralistic ignorance at the end of the group interactions, ending in the
state $s_{i}=1$ and $s_{e}=1$, does not mean that they will readily express
their internal state in the next group interaction because at the beginning
there is always the huge group silence exerting social pressure.

Pluralistic ignorance arises from the complex interplay between individuals’
hidden thoughts and feelings and the externally visible behaviors that are
widely recognized within a group. These visible behaviors are often perceived
as normative, more specifically, they represent descriptive norms [51, 11].
Descriptive norms encompass the perception or understanding of what behaviors
are typically practiced or approved within a specific group or society. They
reflect observations or beliefs about common actions in a given situation,
influencing individuals’ behavior as they seek to conform to perceived
societal standards. The descriptive norm is enforced in our model by the
majority imitation rule. Using the classroom example, the students initially
see their partners not raising their hands and believe that, because the
majority is behaving in this way, that this is prevalent behavior in the
group.

The phenomenon of pluralistic ignorance can be the consequence of behavioral
responses characterized by inaction if this is interpreted as a cue for the
group’s normative behavior. This notion is epitomized by Plato’s famous
saying, “silence gives consent”, highlighting that the absence of behavior can
leave marks in the world. In situations where the internal states of the
others are readily observable, pluralistic ignorance cannot be present. For
example, if there was a neurological mechanism causing individuals to
consistently frown when in doubt, students would immediately recognize their
peers’ lack of comprehension of a subject. We should contrast this mechanism
to the alcohol drinking case where the normative behavior is the action of
drinking. One might be tempted to say that we could use our model to
understand the drinking behavior by interpreting $s_{e}=0$ as the drinking
behavior and $S_{1}=1$ as an internal attitude against abuse of alcohol
consumption. However, at the very beginning of a group interaction (in this
example it would be a campus party), the real behavior would be no drinking at
all, which does not match the social norm saying that excessive drinking is
cool. To models this specific instance of pluralistic ignorance, we would need
to add the social norm to the model explicitly.

There is a similar phenomenon called bystander effect, or bystander apathy,
that is closely linked to pluralistic ignorance and can also be understood
using our results. The bystander effect is a social phenomenon where
individuals are less likely to offer help or intervene in an emergency
situation when other people are present. This effect suggests that the
presence of others can inhibit individual responsibility and action, leading
to a diffusion of responsibility among the bystanders [52]. This phenomenon is
often associated with pluralistic ignorance. At the beginning of the group
interaction, people may want to provide help. However, as they see that nobody
is helping they start to believe that they are the only ones that think so.
This is an example of how failing to overcome pluralistic ignorance can lead
to profoundly negative consequences.

Internalization is often defined as the incorporation of a norm to one’s
personality in such a way that the agent acts accordingly even in the absence
of social pressure [53]. Here we do not model the norm explicitly. Instead,
the norm is inferred from the initial state of the group and expressed in the
tendency to imitate the majority. Using the classroom example, one would say
that “because everyone is showing 0, this must be the group standard”. More
specifically, we adopt the concept of internalization as defined by Bem’s
self-perception theory, which states that individuals subconsciously adjust
their internal beliefs to align with their stated opinions [54].

In summary, pluralistic ignorance is a group phenomenon with potentially
adverse consequences. Instances such as public concern about climate change,
racial segregation, and exaggerated alcohol consumption exemplify scenarios
where individuals’ perceptions of others diverge from reality, resulting in
detrimental outcomes for all involved. Whenever the absence of a behavioral
expression that is under normative constraint matches the normative status of
the activity (for example, initially not raising the hands matches the
normative behavior of not asking questions), pluralistic ignorance can emerge
as a consequence of social pressure to conform.

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#  Supersymmetry of $\mathcal{PT}$\- symmetric tridiagonal Hamiltonians

Mohammad Walid AlMasri<EMAIL_ADDRESS>Department of Physics, Koç
University, Rumelifeneri Yolu, 34450 Sarıyer, Istanbul, Turkey

###### Abstract

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of
non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find
the relation between matrix elements of the non-Hermitian Hamiltonian $H$ and
its supersymmetric partner $H^{+}$ in a given basis. Moreover, the orthogonal
polynomials in the eigenstate expansion problem attached to $H^{+}$ can be
recovered from those polynomials arising from the same problem for $H$ with
the help of kernel polynomials. Besides its generality, the developed
formalism in this work is a natural home for using the numerically powerful
Gauss quadrature techniques in probing the nature of some physical quantities
such as the energy spectrum of $\mathcal{PT}$-symmetric complex potentials.
Finally, we solve the shifted $\mathcal{PT}$-symmetric Morse oscillator
exactly in the tridiagonal representation.

##  I Introduction

Supersymmetric quantum mechanics (SUSY-QM) plays an important rule in the
study of exactly solvable potentials and its properties mainly due to its
powerful factorization techniques which could be used in constructing new
analytically solvable potential Witten ; Krive ; Haymaker ; Cooper2001 ;
Bagchi ; Infeld ; Dong . The relation between one-dimensional solvable
potentials and SUSY was addressed by Gendenshtein through the concept of
shape-invariant potentials Gend . By definition the potential is said to be
shape invariant if its supersymmetric partner has the same spatial dependence
as the original potential with possibly adjusted parameters.

It has been known for a long time that unperturbed partial-wave kinetic energy
operator $H^{0}=-\frac{1}{2}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)}{r^{2}}$
has a tridiagonal representation in a complete Slater or oscillator basis set
$\\{|\phi_{n}\rangle\\}$ Fishman . This was extended later to the case of
Coulomb, Morse and Rosen-Morse potentials haidari . The Supersymmetry of
tridiagonal Hamiltonians was studied in Yamani . It was shown that if a
positive semi-definite Hamiltonian has a tridiagonal representation in a given
orthonormal basis $\\{|\phi_{n}\rangle\\}$, its supersymmetric partner
Hamiltonian $H^{+}$ will have the same matrix representation in the same basis
$\\{|\phi_{n}\rangle\\}$Yamani . With the help of kernel polynomials, the
orthogonal polynomials in the eigenstate expansion problem attached to $H^{+}$
can be recovered from those polynomials arising from the same problem for $H$.
This method was elaborated to solve exactly the supersymmetry of Morse
oscillator Yamani2 .

All previous studies were done with Hermitian tridiagonal Hamiltonians,
however it is known that special class of non-Hermitian Hamiltonians may
posses real eigenvalues Bessis ; Bender ; jones ; Bender1 ; Bender2 . One
feature of these Hamiltonians is that they are all $\mathcal{PT}$-symmetric,
note that the opposite of this statement is not true in general since not all
$\mathcal{PT}$\- symmetric Hamiltonians have real eigenvalues in their energy
spectrum. These observations lead to enormous studies of
$\mathcal{PT}$-symmetric Hamiltonians and their properties especially in
opticsOptic ; Musslimani ; Ramezani ; Longhi . In Jin , a relation between
non-Hermitian systems with the Hermitian scattering systems was found. More
recently Zhang et al. proposed a non-Hermitian supersymmetric array with high-
order exceptional point of arbitrary order and investigated the topological
properties of exceptional point Jin1 . Possible extension to the ideas of
$\mathcal{PT}$-symmetric Hamiltonian would be found in some magnetic systems
such as quantum spin chains and $\mathcal{PT}$-symmetric magnonic waveguides
Lin2 ; Berakdar1 . The supersymmetric quantum mechanical treatment of
$\mathcal{PT}$\- symmetric Hamiltonians was done by Znojil et al Znojil1 and
the supersymmetric method for complex potentials was investigated in Mallik
and can be read from the general chapter by Levai Levai3 .

Throughout the present work, we study the supersymmetry of tridiagonal non-
Hermitian Hamiltonians with real or complex conjugate pair eigenvalues. We
formulate the problem using pseudo-Hermiticity notion where the Hilbert space
of quantum states is endowed with a Hermitian indefinite inner product. Very
recently, supersymmetry of tridiagonal non-Hermitian Hamiltonians was studied
in tri . However, our formalism is different from that presented in tri since
it depends on the Askey scheme of classical orthogonal polynomials. Also we
considered cases where energy spectrum can have discrete and continuous values
simultaneously unlike tri where the energy outcomes where discrete (shifted
harmonic oscillator and squeezed Hamiltonian ). The main motivation of this
work is to develop the notion of $J$-matrix method for complex potentials and
study the supersymmetry of these complex potentials in the tridiagonal
representation. Finally we tested the formalism by solving the
$\mathcal{PT}$-symmetric Morse potential exactly in the tridiagonal
representation domain with the help of orthogonal polynomials.
The organization of paper goes as follows, in section (II) we give a formal
definition of supersymmetric quantum mechanics and its pseudo-Hermitian
generalization . We briefly discuss the hierarchy method of
$\mathcal{PT}$-symmetric tridiagonal Hamiltonians in section (III) . In
section (IV), we express the non-Hermitian Hamiltonians in tridiagonal
representation and calculate the matrix elements for $H$ and its superpartner
$H^{(+)}$. The eigenstates expansion in terms of orthogonal polynomials are
presented in (V). Finally we apply the developed formalism to the case of
$\mathcal{PT}$-symmetric shifted Morse oscillator.

##  II Supersymmnetric Quantum Mechanics and Pseudo-Hermiticity

Supersymmetry combines the bosonic and fermionic degrees of freedom in a
unified algebra. It was first proposed in the context of unification of
particle physics modelsLikhtman ; Ramond ; Wess . However it was realized
later that some ideas of supersymmetry could be used in solving
nonrelativistic quantum mechanical problemsWitten . In SUSY-QM, the Hilbert
space consists of bosonic $\mathcal{H}_{+}$ and fermionic $\mathcal{H}_{-}$
sub-Hilbert spaces i.e. $\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$ for
a given Hamiltonian $H$.

We define the non-Hermitian supersymmetric charges or generators $Q^{I}$,
$I=1,\dots N$ as

$\displaystyle Q^{I}(\mathcal{H}_{\pm})=\mathcal{H}_{\mp},$ (1) $\displaystyle
Q^{I\;\dagger}(\mathcal{H}_{\pm})=\mathcal{H}_{\mp},$ (2)

where $\\{,\\}$ is the anti-commutator. These generators satisfy the following
super-algebra conditions

$\\{Q^{I},Q^{J\;\dagger}\\}=2\delta^{IJ}\;H,$ (3)
$\\{Q^{I},Q^{I}\\}=\\{Q^{J\;\dagger},Q^{J\;\dagger}\\}=0,$ (4)

where $I,J=1,\dots N$ and $\delta^{IJ}$ is the Kronecker delta.
The equations 3 and 4 can also be rephrased in the form

$\displaystyle H=\frac{1}{2}(Q^{I}Q^{I\;\dagger}+Q^{I\;\dagger}Q^{I}),$ (5)
$\displaystyle(Q^{I})^{2}=(Q^{I\;\dagger})^{2}=0,$ (6)

For all $I=1,\dots N$.
To simplify the formalism we write the $2N$-Hermitian SUSY charges as

$\displaystyle Q^{I}_{1}:=\frac{1}{\sqrt{2}}(Q^{I}+Q^{I\;\dagger}),$ (7)
$\displaystyle Q^{I}_{2}:=\frac{1}{\sqrt{2}}(Q^{I}-Q^{I\;\dagger}).$ (8)

and the anti-commutators are

$\displaystyle\\{Q^{I}_{\alpha},Q^{J}_{\beta}\\}=2\delta_{\alpha\beta}\delta^{IJ}H,$
(9) $\displaystyle\\{Q^{I}_{1},Q^{I}_{2}\\}=0,$ (10)

where $\alpha,\beta=1,2$ and $I,J=1,2,\dots N$. Alternatively, the Hamiltonian
takes the form

$H=(Q^{I}_{1})^{2}=(Q^{I}_{2})^{2}$ (11)

In this work we concentrate on $\mathcal{PT}$-symmetric SUSY-QM Znojil1 . By
definition, a $\mathcal{PT}$-symmetric Hamiltonian satisfies

$\mathcal{PT}H(\mathcal{PT})^{-1}=\mathcal{PT}H\mathcal{PT}=H,$ (12)

Where $\mathcal{P}$ and $\mathcal{T}$ are the parity and time-reversal
operators respectively. These operators are defined in the following way

$\mathcal{P}\;x\;\mathcal{P}=-x\;\;,\;\mathcal{P}\;p\;\mathcal{P}=-p\;\;\;,\mathcal{T}i\;\mathbb{I}\mathcal{T}=-i\mathbb{I}.$
(13)

In the case of $\mathcal{PT}$-symmetric Hamiltonians, the inner product of two
states $\phi_{1}$ and $\phi_{2}$ is defined in the Hilbert space $\mathcal{H}$
as Bender

$\langle\langle\phi_{1}|\phi_{2}\rangle\rangle=\langle\phi_{1}|\mathcal{P}|\phi_{2}\rangle.$
(14)

which reduces to the ordinary Hermitian inner product when $\mathcal{P}=1$. In
is worthy to note that $\\{\mathcal{P},Q\\}=0$. The inner product defined in
14 is invariant under time translation generated by the Hamiltonian $H$ if and
only if $H$ is $\mathcal{P}$ pseudo-Hermitian as shown in Mostafazadeh . The
pseudo-superalgebra is given by Znojil1 ( we omitted the index $I$ for the
sake of simplicity)

$\displaystyle\\{Q,\tilde{Q}\\}=2H,\;\;\;Q^{2}=\tilde{Q^{2}}=0,$ (15)

Where $\tilde{Q}=\mathcal{P}^{-1}Q^{\dagger}\mathcal{P}$.
The Hamiltonian $H_{+}=BA$ and $H_{-}=AB$ can be constructed from the shifting
operator $A:\mathcal{H}_{+}\rightarrow\mathcal{H}_{-}$ which maps the
eigenvectors of $H_{+}$ to those of $H_{-}$ while $B$ does the opposite
action.

##  III Factorization of $\mathcal{PT}$-symmetric tridiagonal Hamiltonians

One advantage of supersymmetric quantum mechanics is the factorization
techniques that allow us to factorize Schrödinger equation by expressing the
Hamiltonian as $H=A^{\dagger}A$\+ constant term(s) Cooper2001 ; Sukumar ;
Levai3 . The generic Hamiltonian with tridiagonal representation reads

$H=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)}{2r^{2}}+V_{1}(r)$
(16)

so that
$V^{eff}_{1}=\frac{\hbar^{2}}{2m}\frac{\Psi_{0}^{\prime\prime}}{\Psi_{0}}$
where ′ denotes the derivative with respect to the radial coordinate $r$ and
$V^{eff}_{1}=\frac{\ell(\ell+1)}{2r^{2}}+V_{1}$ is the centrifugal effective
potential which appears usually in three-dimensional quantum problems with
spherical symmetry. It can be defined using the superpotential
$W=-\frac{\hbar}{\sqrt{2m}}[\frac{d\mathrm{ln}\Psi_{0}}{dr}]$ as Cooper2001

$V^{eff}_{1}=W^{2}-\frac{\hbar}{\sqrt{2m}}\frac{dW}{dr}$ (17)

The ground state wavefunction can be computed from the superpotential up to a
normalization constant as

$\Psi_{0}\sim exp[-\frac{\sqrt{2m}}{\hbar}\int^{r}W(r^{\prime})dr^{\prime}]$
(18)

The Hamiltonian 16 factorizes as

$\displaystyle H=A^{\dagger}A$ (19) $\displaystyle H^{\dagger}=AA^{\dagger}$
(20)

Where the bosonic operator $A$ and its Hermitian conjugate are

$\displaystyle
A=\frac{\hbar}{\sqrt{2m}}\frac{d}{dr}+W(r),\;\;A^{\dagger}=-\frac{\hbar}{\sqrt{2m}}\frac{d}{dr}+W(r).$
(21)

For unbroken supersymmetry, the ground state energy is zero. Thus , we have

$H_{1}\Psi_{0}=\frac{-\hbar^{2}}{2m}\frac{d^{2}\Psi_{0}}{dr^{2}}+\frac{\ell(\ell+1)}{2r^{2}}\Psi_{0}+V_{1}\Psi_{0}=0$
(22)

We can solve the differential equation 22 and find an expression for
$V^{eff}_{1}$ defined with respect to the ground state wave function and its
second derivative with respect to the radial coordinate $r$,

$\displaystyle
V^{eff}_{1}=\frac{\hbar^{2}}{2m}\frac{\Psi^{{}^{\prime\prime}}_{0}}{\Psi_{0}}$
(23)

The Hamiltonian 22 factorizes as

$\displaystyle H_{1}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+V_{1}^{eff}(r)$
(24) $\displaystyle=A_{1}A^{\dagger}_{1}+E^{(0)}_{1}.$ (25)

After some calculations we get the Riccati equation for the superpotential
$W_{1}$,

$W^{2}_{1}-W^{\prime}_{1}=\frac{2m}{\hbar^{2}}(V^{eff}_{1}(r)-E^{0}_{1}).$
(26)

The supersymmetric partner of Hamiltonian $H_{1}$ is

$\displaystyle H_{2}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+V_{2}^{eff}(r)$
(27) $\displaystyle=A^{\dagger}_{2}A_{2}+E^{(0)}_{2}.$ (28)

and the corresponding Riccati equation takes the form

$W^{2}_{2}+W^{\prime}_{2}=\frac{2m}{\hbar^{2}}(V^{eff}_{2}(r)-E^{(0)}_{2}).$
(29)

By iteration we can express the general Hamiltonian and Riccati equations
respectively as

$\displaystyle H_{n}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+V_{n}^{tot}(r)$
(30) $\displaystyle=A_{n}A^{\dagger}_{n}+E^{(0)}_{n},\;\;n=1,3,5\dots$
$\displaystyle H_{n}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dr^{2}}+V_{n}^{eff}(r)$
(31) $\displaystyle=A^{\dagger}_{n}A_{n}+E^{(0)}_{n},\;\;n=2,4,6\dots$
$\displaystyle
W^{2}_{n}-W^{\prime}_{n}=\frac{2m}{\hbar^{2}}(V^{tot}_{n}(r)-E^{0}_{n}),n=1,3,5\dots$
(32) $\displaystyle
W^{2}_{n}+W^{\prime}_{n}=\frac{2m}{\hbar^{2}}(V^{eff}_{n}(r)-E^{0}_{n}),n=2,4,6\dots$
(33)

For unbroken supersymmetry, the partner Hamiltonian obey the following
expressions

$E^{0}_{n+1}=E^{1}_{n}$ (34)

with $E^{0}_{0}=0$ and $n=0,1,2,\dots$.

It is not difficult to notice that such procedure should be modified for
complex potentials with $\mathcal{PT}$-symmetry since $V^{eff}$ and $W$ are
complex and should be decomposed to its real and imaginary parts Mallik .
In order to factorize any $\mathcal{PT}$-symmetric Hamiltonians, we introduce
the following four related operators Znojil1

$\displaystyle
A^{(\pm)}=\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+W^{(\pm)}(x)\;\;\;,\;B^{(\pm)}=-\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+W^{(\pm)}(x)$
(35)

and the corresponding Riccati equation is

$H^{(\pm)}=B^{(\pm)}A^{(\pm)}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+[W^{(\pm)}(x)]^{2}-\frac{\hbar}{\sqrt{2m}}[W^{(\pm)}(x)]^{\prime}$
(36)

where $\prime$ is the derivative with respect to the coordinate $x$. In the
incoming sections we will choose the $+$ operators only since the second case
follows the same procedures. The term $\frac{\ell(\ell+1)}{2r^{2}}$ appears in
the three-dimensional problems such as the radial Coulomb potential and
isotropic harmonic oscillator Haymaker .

##  IV J-matrix method for complex tridiagonal Hamiltonians

In ordinary quantum mechanics, it is customary to expand the Hamiltonian in a
basis that makes it diagonal. This simplifies the process of computing the
energy eigenvalues especially for large-sized Hamiltonian matrices. On the
other hand, this approach reduces the class of possible solutions of the
corresponding wave equation of the system. One possible extension of the space
of solutions is known as J-matrix method which assumes the matrix
representation of the Hamiltonian to be tridiagonal in a given orthogonal
polynomial basis Fishman ; haidari ; Mourad . In the J-matrix method, we
expand the wavefunctions over a complete set of square-integrable functions
$L^{2}(\mathbb{R})$ in configuration space
$|\psi(E,x)\rangle=\sum_{n}f_{n}(E)|\phi_{n}(x)\rangle$. The coefficients are
some functions of energy. They encode all physical information about the
system. The wave equation $H|\psi\rangle=E|\psi\rangle$ becomes

$\sum_{n}f_{n}H|\psi\rangle=E\sum_{n}f_{n}|\phi_{n}\rangle$ (37)

By projecting $\langle\phi_{m}$ from left we find

$\sum_{n}f_{n}\langle\phi_{m}|H|\phi_{n}\rangle=E\sum_{n}f_{n}\langle\phi_{m}|\phi_{n}\rangle$
(38)

which can written in matrix from as
$\sum_{n}H_{mn}f_{n}=E\sum_{n}\Omega_{mn}f_{n}$ where
$\Omega_{mn}=\langle\phi_{m}|\phi_{n}\rangle$ is the overlap identity matrix
of the basis elements. Our main task in this approach is to calculate the
matrix elements of the tridiagonal and symmetric operator $J_{mn}$

$J_{mn}=H_{mn}-E\Omega_{mn}=(a_{n}-z)\delta_{n,m}+b_{n}\delta_{n,m-1}+b_{n-1}\delta_{n,m+1}$
(39)

In this work we extend J-matrix method to complex potentials with
$\mathcal{PT}$ symmetry. For this reason we assume the matrix $J_{mn}$ to be
non-self adjoint so that the relation 39 will be written generally as

$J_{mn}=H_{mn}-\varepsilon\Omega_{mn}=(a_{n}-z)\delta_{n,m}+b^{\star}_{n}\delta_{n,m-1}+b_{n-1}\delta_{n,m+1}$
(40)

where in this case the coefficients $a_{n}$ and $b_{n}$ are complex quantities
in general. For a given non-self adjoint Hamiltonian with one-dimensional
complex potential $V$, the reality of its energy bound states requires
$[\mathcal{PT},H]=0$ and $(V(-x))^{\star}=V(x)$ Levai1

The radial Coulomb potential with imaginary charge $iz$ in the atomic units
$\hbar=m=1$

$H=-\frac{1}{2}\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)}{2r^{2}}+\frac{iz}{r}$
(41)

The Hamiltonian 41 has a tridiagonal matrix representation in the non-
orthogonal Laguerre basis Fishman

$\displaystyle\phi_{n}(x)=B_{n}(\lambda r)^{\ell+1}e^{-\frac{\lambda
r}{2}}L^{2\ell+1}_{n}(\lambda r),\;\;n=0,1,2\dots$ (42) $\displaystyle
B_{n}=\sqrt{\frac{\lambda n!}{\Gamma(n+2\ell+2)}}$

Here $\lambda$ is a scale parameter. If we define the $J$-matrix as
$J=(H-\varepsilon)$ , then the only non-zero elements of $J$ are

$\displaystyle(J)_{n,n}=\langle\langle\phi_{n}|H-\varepsilon|\phi_{n}\rangle\rangle=i\lambda
z-(\varepsilon-\frac{\lambda^{2}}{8})(2n+2\ell+2)$ (43)
$\displaystyle(J)_{n,n+1}=\langle\langle\phi_{n}|H-\varepsilon|\phi_{n+1}\rangle\rangle=(\varepsilon+\frac{\lambda^{2}}{8})\sqrt{(n+1)(2n+2\ell+2)}$
(44)

The discrete spectrum can be deduced by imposing the conditions given in
haidari for specific cases,which apply here, and by the procedure explained in
Yamani2 in the general case. Here, we first find the result of demanding that
$J_{\mu,\mu+1}=0$ and then use the result in the requirement that
$J_{\mu,\mu}=0$. These conditions finally give

$\displaystyle(\lambda)_{\mu}=\frac{2iz}{\mu+\ell+1}$ (45)
$\displaystyle(\varepsilon)_{\mu}=\frac{z^{2}}{2(\mu+\ell+1)^{2}}$ (46)

From these results we notice that Coulomb potential with imaginary charge has
positive energy spectrum. Furthermore we notice that there are an infinite
number of positive energy bound-states in the interval
$0\leq(\varepsilon)_{\mu}\leq\frac{z^{2}}{2(\ell+1)^{2}}$ . Also the wave-
function associate with these bound states exhibit sinusoidal behavior as
shown in figure 1. Remarkably, the bound-states exist for both the system with
positive imaginary charge $+iz$ as well as for negative imaginary charge $-iz$
. This stands in contrast to usual Coulomb Hamiltonian which support bound
states only for the case that the charge $z$ is negative. However it is not
$\mathcal{PT}$-symmetric so we did not pursue this direction.

Figure 1: The real and imaginary part of the bound-state wavefunction for the
Coulomb potential with complex charge and different configurations shown above
each plot.

For one-dimensional complex potentials the term $\frac{\ell(\ell+1)}{2r^{2}}$
is absent. We apply the J-matrix method for the case of
$\mathcal{PT}$-symmetric Rosen-Morse II potential Levai which has applications
in optics as shown in Roychoudhury . The $\mathcal{PT}$-symmetric Rosen-Morse
II potential is

$V(x)=2iB\tanh\delta
x-A(A+\frac{\delta\hbar}{\sqrt{2m}})\;\mathrm{sech}^{2}\delta x$ (47)

where $\alpha$ is a free real parameter, $A,B$ are coefficients and $x$ is the
coordinate variable in one-dimension. The corresponding associated energy
eigenvalues are

$E_{n}=-(A-\frac{n\hbar\delta}{\sqrt{2m}})^{2}+\frac{B^{2}}{(A-\frac{n\hbar\delta}{\sqrt{2m}})^{2}}$
(48)

which are real. The basis function for this potential can be written in virtue
of Jacobi polynomials $P^{\mu,\nu}_{n}$ with complex parameters since
$\mu,\nu\in\mathbb{C}$ such that $\mu=\overline{\nu}$

$\psi_{n}(y)=C_{n}(1-y)^{\alpha}(1+y)^{\beta}P^{\mu,\nu}_{n}(y)$ (49)

where $y=\tanh\delta x$ and The parameters $\mu=s-n+a$, $\nu=s-n-a$ where
$s=\frac{\sqrt{2m}A}{\hbar\delta}$, $\lambda=i\frac{\sqrt{2m}B}{\hbar\delta}$
and $a=\frac{\sqrt{2m}\lambda}{\hbar\delta(s-n)}$ are complex conjugate
$\overline{\mu}=\nu$ unlike the real case Levai ; haidari ; Ismail .
Analytical solutions are obtainable only for the following cases
$(\alpha,\beta)=(\frac{s-n+a}{2},\frac{s-n-a}{2}),(\frac{s-n+a+1}{2},\frac{s-n-a}{2}),(\frac{s-n+a}{2},\frac{s-n-a+1}{2})$
haidari .

The generalized Rosen-Morse II

$V_{g}(x)=2iB\tanh\alpha
x-A(A+\frac{\alpha\hbar}{\sqrt{2m}})\mathrm{sech}^{2}\alpha x+iC\tanh\alpha
x\;\mathrm{sech}^{2}\alpha x$ (50)

It has tridiagonal representation in Jacobi polynomial basis
$\psi_{n}(x)=(1-\tanh(\alpha x))^{\frac{\mu}{2}}(1+\tanh(\alpha
x))^{\frac{\nu}{2}}P^{(\mu,\nu)}_{n}(\tanh(\alpha x))$ in the case
$(\alpha,\beta)=(\frac{s-n+a}{2},\frac{s-n-a}{2})$ where
$2iB=(\frac{\delta\mu}{2})^{-}(\frac{\delta\nu}{2})^{2}$.

Another interesting example is the $\mathcal{PT}$-symmetric Scarf II potential
$V_{Scarf}(x)=-V_{1}\;\mathrm{sech}^{2}x+iV_{2}\;\mathrm{sech}x\tanh x$ for
$V_{1}>0$ and $V_{2}\neq 0$ which was found to host an exceptional point in
its energy spectrum (zero-width resonance or spectral singularity in other
references ) for some wavefunction Ahmed ; Ahmed1 ; Quesna . Similar to the
complex Rosen-Morse II potential, the complex Scarf II potential can be solved
exactly in a Jacobi polynomial basis Quesna ; Levai4 .

##  V  Pseudo-Hermitian Supersymmetric Hamiltonians in Tridiagonal
Representation

We assume the matrix representation of a given $\mathcal{PT}$-symmetric
Hamiltonian $H=BA$ in a complete orthonormal basis $\\{|\phi_{n}\rangle\\}$
where $n=0,1,2,\dots$, to be tridiagonal in the following manner

$\langle\langle\phi_{n}|H|\phi_{m}\rangle\rangle=b_{n-1}\;\delta_{n,m+1}+a_{n}\;\delta_{n,m}+b^{\star}_{n}\;\delta_{n,m-1},$
(51)

where $\\{a_{n},b_{n}\\}$ are complex coefficients, and $b_{-1}=0$. We define
the operator $A$ by its action on each element of the basis
$\\{|\phi_{n}\rangle\\}$ as

$A\;|\phi_{n}\rangle=c_{n}\;|\phi_{n}\rangle+d_{n}\;|\phi_{n-1}\rangle.$ (52)

where the coefficients $\\{c_{n},d_{n}\\}$ are to be defined later expect to
$d_{0}=0$. The action of operator $B$ on a given basis
$\\{|\phi_{n}\rangle\\}$ is

$B\;|\phi_{n}\rangle=c^{\star}_{n}\;|\phi_{n}\rangle+d^{\star}_{n+1}\;|\phi_{n+1}\rangle$
(53)

$n=0,1,\dots$, and $\star$ is the complex conjugation operation. Note that
when $c_{n}=0$ in equations 52 and 53, the operators $A$ and $B$ become the
lowering and raising operators, respectively. The product operator $BA$ has
the following tridiagonal representation in the given basis
$\\{|\phi_{n}\rangle\\}$ as follows

$\displaystyle\langle\langle\phi_{n}|BA|\phi_{m}\rangle\rangle=c_{m}d^{\star}_{m+1}\;\delta_{n,m+1}+(|c_{m}|^{2}+|d_{m}|^{2})\;\delta_{m,n}$
(54) $\displaystyle+d_{m}c^{\star}_{m-1}\;\delta_{n,m-1}.$

where $n,m=1,2,\dots$. We prove this last result by acting with 52 and 53 on a
given basis $\\{|\phi_{n}\rangle\\}$, we obtain

$\displaystyle
BA|\phi_{m}\rangle=B(A|\phi_{m}\rangle)=B(c_{m}\;|\phi_{m}\rangle+d_{m}\;|\phi_{m-1}\rangle)$
(55)
$\displaystyle=c_{m}c^{\star}_{m}\;|\phi_{m}\rangle+c_{m}d^{\star}_{m+1}\;|\phi_{m+1}\rangle+d_{m}c^{\star}_{m-1}\;|\phi_{m-1}\rangle+d_{m}d^{\star}_{m}\;|\phi_{m}\rangle$
$\displaystyle=c_{m}d^{\star}_{m+1}\;|\phi_{m+1}\rangle+(|c_{m}|^{2}+|d_{m}|^{2})\;|\phi_{m}\rangle+d_{m}c^{\star}_{m-1}\;|\phi_{m-1}\rangle$

Next, the matrix elements $\langle\langle\phi_{m}|BA|\phi_{n}\rangle\rangle$
can be calculated by taking the inner product of the previous relation
alongside with the orthonormality condition
$\langle\langle\phi_{m}|\phi_{n}\rangle\rangle=\langle\phi_{m}|\mathcal{P}|\phi_{n}\rangle=\delta_{m,n}$.
This gives the desired result

$\displaystyle\langle\langle\phi_{m}|BA|\phi_{n}\rangle\rangle=c_{n}d^{\star}_{n+1}\;\delta_{m,n+1}+(|c_{n}|^{2}+|d_{n}|^{2})\;\delta_{m,n}+d_{n}c^{\star}_{n-1}\;\delta_{m,n-1}.$
(56)

The coefficients $\\{a_{n},b_{n}\\}$ defined in 51 are connected with
$\\{c_{n},d_{n}\\}$ via the relations

$\displaystyle a_{n}=|c_{n}|^{2}+|d_{n}|^{2};$ (57) $\displaystyle
b_{n}=c_{n}\;d^{\star}_{n+1}$ (58)

From 57 and 58 , we find the following condition between the coefficients

$a_{0}d^{\star}_{1}=|b_{0}|^{2}.$ (59)

Now, let us calculate the supersymmetric partner Hamiltonian $H^{+}$ in the
same basis $\\{|\phi_{m}\rangle\\}$ of the original Hamiltonian. It is not
difficult to notice that $H^{+}$ has a tridiagonal representation in the same
basis of $H$ as follows

$\langle\langle\phi_{m}|H^{+}|\phi_{n}\rangle\rangle=b^{+}_{n-1}\;\delta_{n,m+1}+a^{+}_{n}\;\delta_{n,m}+(b^{\star}_{n})^{+}\;\delta_{n,m-1},$
(60)

If the operators $A$ and $B$ act on the basis elements
$\\{|\phi_{n}\rangle\\}$ according to 52 and 53 respectively. The product
operator $AB$ has the following tridiagonal representation in the given basis
$\\{|\phi_{n}\rangle\\}$ as follows:

$\displaystyle\langle\langle\phi_{n}|AB|\phi_{m}\rangle\rangle=d_{m+1}^{\star}c_{m+1}\;\delta_{n,m+1}+(|c_{m}|^{2}+|d_{m+1}|^{2})\;\delta_{m,n}+$
(61) $\displaystyle c^{\star}_{m}d_{m}\;\delta_{n,m-1}.$

where $n,m=1,2,\dots$. The coefficients in 60 are connected with $c_{n}$ and
$d_{n}$ in the following way

$\displaystyle a_{n}^{+}=|c_{n}|^{2}+|d_{n+1}|^{2},$ (62) $\displaystyle
b_{n}^{+}=d^{\star}_{n+1}c_{n+1}.$ (63)

If we consider $A$ and $B$ as lowering and raising operators respectively, we
would then have vanishing $c_{n}$ and $c^{\star}_{n}$ coefficients. In that
case, both $H$ and $H^{(+)}$ are diagonal in this basis
$\\{|\phi_{n}\rangle\\}$ as follows

$\displaystyle H_{nm}=|d_{n}|^{2}\delta_{m,n},$ (64) $\displaystyle
H^{(+)}_{nm}=|d_{n+1}|^{2}\delta_{m,n}.$

As a final comment, the energy eigenvalues $E^{(+)}_{n}$ of the superpartner
Hamiltonian $H^{(+)}$ are connected with the energy eigenvalues $E_{n}$ of the
Hamiltonian $H$ through the relation $E^{(+)}_{n}=E_{n+1}$ as expected for any
SUSY quantum mechanical model.

##  VI Eigenstates expansion using orthogonal polynomials

The action of Hamiltonian $H$ in the tridiagonal representation is

$H|\phi_{n}\rangle=b_{n-1}|\phi_{n-1}\rangle+a_{n}|\phi_{n}\rangle+b^{\star}_{n}|\phi_{n+1}\rangle$
(65)

$n=0,1,\dots$. One other hand, the Hamiltonian $H$ obeys the following
eigenvalue equation

$H|\psi_{E}\rangle=E|\psi_{E}\rangle,$ (66)

where $|\psi_{E}\rangle$ is the set of energy eigenstates. It is convenient in
our case to expand $|\psi_{E}\rangle$ in term of the basis set
$\\{|\phi_{n}\rangle\\}$ as

$|\psi_{E}\rangle=\Sigma_{n=0}^{\infty}\;C_{n}(E)\;|\phi_{n}\rangle.$ (67)

Here $C_{n}(E)$ are defined as complex coefficients in general. Then, making
use of equation 67 and the orthonormality of basis set $\\{|\phi_{n}\\}$ in 65
gives the following recurrence relation of the expansion coefficients $C(E)$

$E\;C_{n}(E)=b_{n-1}\;C_{n-1}(E)+a_{n}\;C_{n}(E)+b^{\star}_{n}\;C_{n+1}(E).$
(68)

The coefficients $C_{n}(E)$ satisfy the following orthonormality condition in
case of discrete energy spectrum

$\displaystyle\int_{\Omega(E)}C^{\star}_{n}(E)\;C_{m}(E)\;dE=\int_{\Omega(E)}\langle\langle\phi_{n}|\psi_{E}\rangle\rangle\langle\langle\psi_{E}|\phi_{m}\rangle\rangle\;dE=\langle\langle\phi_{n}|\phi_{m}\rangle\rangle=\delta_{n,m}.$
(69)

where $\Omega(E)$ is the energy support of the Hamiltonian $H$. For general
case, when we have both discrete and continuous energy spectrum, the
orthonormality condition becomes

$\delta_{m,n}=\Sigma_{\alpha}\;C_{\alpha}^{\star}(E_{\alpha})\;C_{m}(E_{\alpha})+\int_{\Omega(E)}C^{\star}_{n}(E)\;C_{m}(E)\;dE.$
(70)

Let us define $P(E)$ as

$P_{n}(E)=\frac{C_{n}(E)}{C_{0}(E)}$ (71)

with $P_{0}(E)=1$ and $P_{1}(E)=(E-a_{0})(b^{\star}_{0})^{-1}$. By plugging 71
in 68 we obtain the following three-recursion formula

$EP_{n}(E)=b_{n-1}P_{n-1}(E)+a_{n}P_{n}(E)+b^{\star}_{n}P_{n+1}(E).$ (72)

The coefficients $(c_{n},d_{n})$ are related to the values at zero of
consecutive polynomial $P_{n}(0)$ as

$\displaystyle(d^{\star}_{n+1})^{2}=-b_{n}\;\frac{P_{n}(0)}{P_{n+1}(0)},$ (73)
$\displaystyle|c_{n}|^{2}=-b^{\star}_{n}\;\frac{P_{n+1}(0)}{P_{n}(0)}.$ (74)

Since the supersymmetric partner Hamiltonian has the same tridiagonal
representation in the same basis, we write

$EP_{n}^{(+)}(E)=b_{n-1}^{(+)}\;P_{n-1}^{(+)}+a^{(+)}_{n}\;\
P^{(+)}_{n}(E)+(b^{(+)}_{n})^{\star}\;P^{(+)}_{n+1}(E).$ (75)

With the initial conditions $P^{(+)}_{0}(E)=1$ and
$P^{(+)}_{1}(E)=\frac{(E-a_{0}^{+})}{({b}_{0}^{(+)})^{\star}}$. The
polynomials $P_{n}(E)$ are related to their supersymmetric partner
$P^{(+)}_{n}(E)$ via the following relation

$P^{(+)}_{n}(E)=\sqrt{\frac{b_{0}P_{1}(0)}{b_{n}P_{n}(0)P_{n+1}(0)}}\;K_{0}(E,0),$
(76)

where $K_{n}(E,0)=\Sigma_{j=0}^{n}P_{j}(E)P_{j}(0)$ denotes the kernel
polynomials Yamani .

##  VII Supersymmetry of non-Hermitian $\mathcal{PT}$-symmetric Morse
oscillator

The generalized non-Hermitian $\mathcal{PT}$-symmetric Morse potential is
Morse ; Znojil ; Quesna1 ; Quesna2

$\displaystyle V(x)=V_{1}e^{-2i\alpha x}-V_{2}e^{-i\alpha x}$ (77)
$\displaystyle=V(e^{-2i\alpha x}-qe^{-i\alpha x}).$ (78)

where $V=V_{1}$ and $q=\frac{V_{2}}{V_{1}}$.
We choose $V=V_{0}$ and $q=2$ , then the potential reads

$V(x)=V_{0}(e^{-2i\alpha x}-2e^{-i\alpha x}),$ (79)

where $V_{0}$ and $\alpha$ are real parameters of the potential. One
interesting feature of the Morse Hamiltonian is the possibility of having a
discrete energy spectrum depending on the parameter $V_{0}$ besides its
continuous ones. However, in contrast to real case, the potential 79 is
bounded and periodic function for fixed values of $V_{0}$. The total
Hamiltonian $H$ is unbounded non-self adjoint operator. Moreover, it has
imaginary and real parts. Another alternative way for obtaining real
eigenvalues of energy was addressed in Znojil for generalized Morse potential
defined in 77 . We apply our developed formalism in earlier sections to the
case of $\mathcal{PT}$-symmetric shifted Morse Hamiltonian written in atomic
units $\hbar=m=1$ as Yamani2

$H=-\frac{1}{2}\;\frac{d^{2}}{dx^{2}}+V_{0}\;(e^{-2i\alpha x}-2e^{-i\alpha
x})-\frac{1}{2}\alpha^{2}D^{2},$ (80)

where $D=\sqrt{\frac{-2V_{0}}{\alpha^{2}}}-\frac{1}{2}$.

Before we start our analysis it would be important to trace the energy
spectrum of the $\mathcal{PT}$-symmetric Morse potential and verify it is real
on some interval. To answer this exactly we numerically find the eigenvalues
of 79 in a Hermite orthogonal basis endowed with a scale parameter $\lambda$

$\psi_{n}(x)=A_{n}e^{-\frac{\lambda^{2}x^{2}}{2}}H_{n}(\lambda x),$ (81)

where $A_{n}=\sqrt{\frac{\lambda}{2^{n}n!\sqrt{\pi}}}$. This basis does not
make 79 tridiagonal but it is nice on two counts. First it is complete in the
$x$-space over the interval $(-\infty,\infty)$ which is the range of Morse
potential. Second, for any desired value $N$, it generates two sets of numbers
, the discrete points $\\{x_{\mu}\\}_{\mu}^{N-1}$ and the associated matrix
$\\{\Gamma_{n,\mu}\\}_{\mu,n}^{N-1}$ to enable us doing integrals
approximately, but accurately, simply as a quadrature

$S_{n,m}\equiv\int
dx\;\psi_{n}(x)S(x)\psi_{n}(x)\simeq\Sigma_{\mu=0}^{N-1}\;\Gamma_{n,\mu}\;S(x_{\mu})\Gamma_{m,\mu}.$
(82)

For any function $S(x)$ such as $\mathcal{PT}$-symmetric Morse potential. For
$N=70$ and $\lambda=12$ we were able to verify the reality of eigenvalues for
the $\mathcal{PT}$-symmetric Morse potential. For lower scale parameter some
complex eigenvalues creep usually in pairs as shown in Figure 1 and 2. If we
go higher than $\lambda=10$, we have a pure real spectrum. The Gauss
quadrature treatment can be used for all complex potentials. It is convenient
in this case to consider orthogonal polynomials with three-recurrence relation
on the interval of studyGolub ; Broad ; Reinhardt .

Figure 2: The imaginary part of the $\mathcal{PT}$-symmetric Morse potential
with respect to the real part with $\lambda=1.5$ and $N=70$. We notice that
complex (also the real) eigenvalues creep usually in pairs. Figure 3: The
imaginary part of the $\mathcal{PT}$-symmetric Morse potential with respect to
the real part with $\lambda=10$ and $N=70$. We get a totally real spectrum
except for one pair of complex energy eigenvalues. If we jack up the value of
scale parameter higher than $\lambda=10$ this last pair disappear, and we have
a pure real spectrum.

The complex Morse Hamiltonian has tridiagonal representation using the
following orthogonal basis

$\phi_{n}(x)=\sqrt{\frac{i\alpha
n!}{\Gamma(n+2\gamma+1)}}\;(\xi)^{\gamma+\frac{1}{2}}\;e^{-\frac{1}{2}\xi}\;L_{n}^{2\gamma}(\xi),$
(83)

where $\xi=\sqrt{\frac{-8V_{0}}{\alpha^{2}}}\;e^{-i\alpha x}$ and
$L_{n}^{2\gamma}(\xi)$ is the associated Laguerre polynomialMOS . The
parameter $\gamma$ can take any value except $-\frac{1}{2}$. In contrast to
the real case, $L_{n}^{\mu}(z)$ is a function of complex variable $z$. The
basis 83 vanishes asymptotically when $x\rightarrow\pm\infty$ if and only if
$V_{0}$ is negative Levai1 . We considered one set of basis 83 for simplicity.
However as emphasized in Znojil it is possible to write the total wavefunction
for the one-dimensional $\mathcal{PT}$-symmetric Morse potential as a sum of
two terms one of them is what I considered in 83 and other term which is the
same but with $\gamma\rightarrow-\gamma$. The basis set 83 can be written in
term of confluent hypergeometric function of the first kind $1F_{1}(a;b;x)$
since
$L^{2\gamma}_{n}(\xi)=\frac{(2\gamma+1)_{n}}{n!}1F_{1}(-n;2\gamma+1;\xi)$
where $(2\gamma+1)_{n}$ is the Pochhammer symbol KS .

The matrix representation of the shifted Morse Hamiltonian is

$\langle\langle\phi_{n}|H|\phi_{m}\rangle\rangle=b_{n-1}\;\delta_{n,m+1}+a_{n}\;\delta_{n,m}+b^{\star}_{n}\;\delta_{n,m-1}.$
(84)

Since the Hamiltonian $H$ obeys the eigenvalue equation
$H|\psi_{E}\rangle=E|\psi_{E}\rangle$ we can make the following expansion
$|\psi_{E}\rangle=\Sigma_{n=0}^{\infty}\;C_{n}(E)\;|\phi_{n}\rangle$. We note
that $C(E)$ satisfies a recursion relation similar to 68. More explicitly, we
have the following

$\displaystyle
E\;C_{n}(E)=-\frac{\alpha^{2}}{2}\;\sqrt{n(n+2\gamma)}\big{(}n-\frac{1}{2}+\gamma-D)\;C_{n-1}(E)$
(85)
$\displaystyle+\frac{\alpha^{2}}{2}\;\bigg{(}\big{(}n+\gamma+\frac{1}{2}-D\big{)}^{2}+n(n+2\gamma)-D^{2}\bigg{)}\;C_{n}(E)$
$\displaystyle-\frac{\alpha^{2}}{2}\sqrt{(n+1)(n+2\gamma+1)}\big{(}n+\gamma+\frac{1}{2}-D\big{)}\;C_{n+1}(E).$

In contrast to the real case, $D$ is imaginary for positive $V_{0}$. The
polynomials $\mathcal{C}_{n}(E)$ can be defined in terms of the continuous
dual Hahn polynomials $S_{n}$ as

$\displaystyle
C_{n}(E)=\sqrt{\frac{\Gamma(n+2\gamma+1)}{n!}}\;S_{n}(\lambda^{2};-D,\gamma+\frac{1}{2},\gamma+\frac{1}{2})$
(86)
$\displaystyle=\sqrt{\frac{\Gamma(n+2\gamma+1)}{n!}}\;\lim_{d\rightarrow\infty}\frac{W_{n}(\lambda^{2};-D,\gamma+\frac{1}{2},\gamma+\frac{1}{2},d)}{(-D+d)_{n}},$

where $\lambda=\sqrt{-2\alpha^{-2}E-D^{2}}$ and $W_{n}$ is the Wilson
polynomials with $d$ being an arbitrary parameter KS . From the previous
relation we obtain the following exact expression for $P_{n}$ ,

$\displaystyle P_{n}\left(E\right)=\frac{C_{n}(E)}{C_{0}(E)}$ (87)
$\displaystyle=\frac{\left(-1\right)^{n}\left(\gamma+\frac{1}{2}-D)\right)_{n}}{\sqrt{n!\left(2\gamma+1\right)_{n}}}\times$
$\displaystyle\text{
}_{3}\digamma_{2}\left(\begin{array}[]{ccc}-n&1-D+i\lambda&1-D-i\lambda\\\
&\gamma+\frac{1}{2}-D&\gamma+\frac{1}{2}-D\end{array}\mid 1\right)$ (90)

Following the developed formalism in V, we need to calculate the coefficients
$(c_{n},d_{n})$ in order to find $a^{+}_{n}$ and $b^{+}_{n}$ that appear in
the matrix representation of supersymmetric partner Hamiltonian $H^{+}$ 60.
Thus, we have the following relations

$c_{n}=\frac{i\alpha}{\sqrt{2}}(n+\gamma+\frac{1}{2}-D),$ (91)

$\displaystyle d_{n+1}=-\frac{i\alpha}{\sqrt{2}}\;\sqrt{(n+1)(n+1+2\gamma)}.$
(92)

By plugging 91, 92 in 62 and 63, we find

$a_{n}^{\left(+\right)}=\frac{\alpha^{2}}{2}\left(\left(n+1\right)\left(n+2\gamma+1\right)\right)+\big{(}n+\gamma+\frac{1}{2}-D\big{)}^{2}$
(93) $\mathit{\
}b_{n}^{\left(+\right)}=-\frac{\alpha^{2}}{2}\sqrt{\left(n+1\right)\left(n+2\gamma+1\right)}\left(n+\gamma+\frac{3}{2}-D\right)$
(94)

for every $n=0,1,2,...$ . From 76 we can find the relation for
$P_{n}^{(+)}\left(E\right)$ provided that we have determined
$P_{n}\left(E\right)$ in 87

$\displaystyle
P_{n}^{(+)}\left(E\right)=\frac{\left(-1\right)^{n}\left(\gamma+\frac{3}{2}-D)\right)_{n}}{\sqrt{n!\left(2\gamma+2\right)_{n}}}\times$
(95) $\displaystyle\text{
}_{3}\digamma_{2}\left(\begin{array}[]{ccc}-n&1-D+i\lambda&1-D-i\lambda\\\
&\gamma+\frac{3}{2}-D&\gamma+\frac{3}{2}-D\end{array}\mid 1\right).$ (98)

comparing 95 with 87 we found that the supersymmetric partner of $P_{n}(E)$
can be obtained by the substitution $\gamma\rightarrow\gamma+1$ as expected
for supersymmetric theory.

##  VIII Conclusion

In this work, we have extended the study of supersymmetric tridiagonal
Hamiltonians to include $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians.
Our developed formalism can be used for both Hermitian and non-Hermitian
Hamiltonians so it provides a generalization to Yamani . Moreover, we
developed the notion of $J$-matrix method for complex potentials. We have
calculated the matrix coefficients for the Hamiltonian $H$ and its
supersymmetric partner $H^{(+)}$. In contrast to the Hermitian Hamiltonian
case described in Yamani , we found the coefficients $a_{n}$ and $b_{n}$ in
eq. 51 to be imaginary. Moreover we have expanded the eigenstates of $H$ and
its supersymmetric partner in terms of orthogonal polynomials. We proved that
orthogonal polynomials attached to $H^{(+)}$ can be determined from the
orthogonal polynomials attached to $H$ with the help of kernel polynomials. We
examined our formalism by solving the $\mathcal{PT}$-symmetric shifted Morse
potential exactly. In this case, we found the coefficient polynomials
$P_{n}(E)$ and its supersymmetric partner $P^{+}_{n}(E)$ to be given in term
of the hypergeometric function $\text{ }_{3}\digamma_{2}$. These results make
our task of classifying complex potentials with real energy eigenvalues easier
when one consider the Askey scheme which is based mainly hypergeometric
functions KS .

Acknowledgment. The author is very much indebted to Professor Miloslav Znojil
for his comments on earlier draft of this work. The author would like to thank
the anonymous Referees for the constructive comments which improved the
presentation of this work.

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