# Improving End-of-Turn Detection in Spoken Dialogues by Detecting Speaker
Intentions as a Secondary Task

###### Abstract

This work focuses on the use of acoustic cues for modeling turn-taking in
dyadic spoken dialogues. Previous work has shown that speaker intentions
(e.g., asking a question, uttering a backchannel, etc.) can influence turn-
taking behavior and are good predictors of turn-transitions in spoken
dialogues. However, speaker intentions are not readily available for use by
automated systems at run-time; making it difficult to use this information to
anticipate a turn-transition. To this end, we propose a multi-task neural
approach for predicting turn-transitions and speaker intentions
simultaneously. Our results show that adding the auxiliary task of speaker
intention prediction improves the performance of turn-transition prediction in
spoken dialogues, without relying on additional input features during run-
time.

Index Terms—  Multi-task learning, recurrent neural networks, LSTM, turn-
taking, spoken dialogues, speaker intentions

00footnotetext: ∗Work completed while at IBM T. J. Watson Research Center.

##  1 Introduction

Dialogue agents must be able to engage in human-like conversations in order to
make interactions with spoken dialogue systems more natural and less rigid.
Turn-management is an essential component of conversations as it allows
participants in a dialogue to exchange control of the floor. Studies have
shown that conversation partners rely on both syntactic and prosodic cues to
anticipate turn-transitions [1, 2, 3]. Syntactic cues include keywords and
semantics of an uttered sentence. Prosodic cues include the final intonation
of a clause, pitch level, and speaking rate. In this work, we assess the
efficacy of using acoustic cues for anticipating turn-switches in dyadic
spoken dialogues. Given a single utterance, our goal is to use acoustic cues
to predict if there will be a switch in speakers for the upcoming utterance or
not.

Modern spoken dialogue systems generally rely on simple thresholding
approaches for modeling turn-taking [4, 5, 6]. However, turn-management is a
complex phenomenon, in which participants in a conversation rely on multiple
cues to anticipate turn changes or end-of-turns. We anticipate that
interactions between humans and machines can be improved if dialogue systems
can accurately anticipate turn-switches in spoken conversations.

Turn-taking in conversations can take many forms. The two basic turn-taking
functions are hold and switch. Given an utterance in a conversation, a hold
indicates that the next utterance will be uttered by the same speaker while a
switch indicates that the next utterance will be uttered by the other speaker
in the conversation. Turn-switches can be further divided into smooth and
overlapping switches [3]. Smooth switches occur when there is silence between
two consecutive utterances from two speakers. Overlapping switches occur when
a speaker starts uttering a sentence before the other speaker finishes
uttering his/her sentence.

Previous works built models that used both acoustic and syntactic information
to anticipate turn-changes to help make turn-management more natural in spoken
dialogue systems [3, 5]. Gravano and Hirschberg showed that raising contours
of intonation correlates with turn-transitions while flat intonations
correlates with turn-holds [3]. They also showed that certain keywords (e.g.,
“you know…”) and textual completion have good correlations with turn-
management functions. In addition to the usefulness of syntactic and acoustic
cues for modeling turn-taking, previous work showed that speaker intentions
(e.g., ask a question, utter a backchannel, etc.) can be good predictors of
turn-transitions in dialogues [7, 8]. For example, a switch in speaker turns
is more likely to occur after encountering a question than it is to occur
after encountering a statement. Although speaker intentions (sometimes
referred to as dialogue acts) are useful for predicting turn-transitions [7,
8], they require human annotations and are not readily available during run-
time.

We propose the use of a multi-task Long Short-Term Memory (LSTM) network that
takes in a sequence of acoustic frames from a given utterance and predicts
turn-transitions and speaker intentions simultaneously. During training time,
the network is optimized with a joint loss function using ground-truth labels
for turns and intentions. During test time, the network makes two predictions,
one of which can be discarded or used by other modules in a spoken dialogue
system. The advantage is that this allows the model to use representations
that encode information about speaker intentions for anticipating turns
changes. Our experiments demonstrate that adding the detection of speaker
intentions as a secondary task improves the performance of anticipating turn-
transitions.

##  2 Related Work

The problem of modeling turn-taking in conversations has been extensively
studied in the literature. In this section, we give an overview of related
works that focused on speech or textual interactions (i.e., no visual cues).
Our work complements previous work by showing that a model that uses acoustic
cues for predicting turn-switches benefits from adding speaker intentions
prediction as an auxiliary task using the multi-task learning framework.

One line of work looked at the use of acoustic and lexical features for
modeling turn-taking behavior [4, 5, 9, 10]. Liu et al. [5], Masumura et al.
[9], and Ishimoto et al. [10] looked at the problem in Japanese conversations
while Maier et al. [4] looked at the problem in German conversations. Masumura
et al. proposed using stacked time-asynchronous sequential networks for
detecting end-of-turns given sequences of asynchronous features (e.g., MFCCs
and words) [9]. Ishimoto et al. investigated the dependency between syntactic
and prosodic features and showed that combining the two features is useful for
predicting end-of-turns [10]. Liu et al. built a Recurrent Neural Network
(RNN) to classify a given utterance into four classes that relate to turn-
taking behavior using joint acoustic and lexical embeddings [5]. Finally,
Maier et al. built an LSTM with a threshold-based decoding and studied the
trade-off between latency and cut-in rate for end-of-turn detection in
simulated real-time dialogues [4]. The conclusion reached by this line of work
was that end-of-turn detection models benefit from augmenting classifiers that
use acoustic information with lexical information.

Another line of work focused solely on the acoustic modality, pointing out
that using lexical features would (1) require access to a speech recognition
pipeline and (2) bias the classifiers due to varying prompt types [11].
Arsikere et al. compared the effectiveness of acoustic features (e.g., pitch
trends, spectral constancy, etc.) for predicting end-of-turns in two datasets
that differed in prompt type (one is slow and deliberate, the other is fast
and spontaneous) [11]. They found that the same acoustic cues were useful for
detecting end-of-turns for both prompt types.

A final line of work used dialogue act information when modeling turn-taking
behavior [7, 8, 12]. Guntakandla and Nielsen built a turn-taking model that
relied on transcribed segments, intention labels, speaker information, and
change in speaker information to predict turn-transitions in dialogues [8].
Meshorer and Heeman used current and past speaker intention labels along with
two new features, relative turn length and relative floor control, summarizing
past speaker behavior for predicting turn-switches in dialogues [7]. Finally,
Heeman and Lunsford showed that turn-taking behavior not only depends on
previous and upcoming speech act types, but also depends on the nature of a
dialogue; suggesting that turn-taking events should be split into several
groups depending on speech act types and the context of the dialogue [12].

The works of Meshorer and Heeman [7], Guntakandla and Nielsen [8], and Heeman
and Lunsford [12] suggested that speaker intentions can be useful for
predicting turn-transitions. However, speaker intentions are not readily
obtainable from utterances and require manual human annotations. We are
interested in studying how we can augment acoustic systems with speaker
intention information, available during training time, to improve performance
of turn-transitions predictions.

##  3 Problem Setup

We follow the work of Meshorer and Heeman [7] and represent a conversation
between two speakers as a sequence of utterances, taking the following form:

$u_{1},u_{2},\ldots,u_{N}$

where each $u_{i}$ is an utterance in the conversation. The sequence of
utterances are sorted in terms of start talk time. Let $spkr(\cdot)$ be a
function that returns the speaker of a given utterance. Given $u_{i}$, the
goal is to predict whether the following statement is true or false:

$spkr(u_{i})\neq spkr(u_{i+1})$

If the statement is true, then a turn-switch will take place and the other
speaker will speak next. If the statement is false, then the current speaker
will continue speaking.

Each utterance in the sequence represents a complete sentence, containing both
acoustic and lexical cues, and varies in duration. We assume that we know the
end-points of each utterance as in [7, 8, 11]. Utterance end-points can be
readily obtained from modern voice-activity detection algorithms or using an
end-of-utterance detection systems (e.g., [6]). We leave the problem of
combining our multi-task model with end-of-utterance detection for future work
and focus on the problem of predicting turn-switches from acoustic cues for a
given utterance.

##  4 Dataset and Features

###  4.1 Dataset

We use the Switchboard corpus [13] to model turn-taking behavior in spoken
dialogues. The corpus consists of dyadic telephony conversations between
participants who were asked to discuss various topics. We use the annotations
provided by the Switchboard Dialog Act
Corpus111https://github.com/cgpotts/swda (SwDA), since we are interested in
utilizing speaker intentions (i.e., dialogue act types). The goal of SwDA
corpus was to extend the original Switchboard corpus with dialogue act types
that summarize turn information in the conversations.

However, the SwDA corpus does not map dialogue acts to timing information in
the original media files of the Switchboard corpus. It only maps dialogue acts
to lexical and turn information. We augment the SwDA corpus with the NXT
Switchboard corpus [14] to get utterance timing information from the original
media files. The aim of the NXT Switchboard corpus was to combine major
annotations that were performed on the Switchboard corpus and make them
accessible within one framework.

Preparation. We first add binary turn labels (switch/hold) to each utterance
in the dataset. We focus on $7$ major dialogue acts which we obtain by
grouping different SwDA classes as shown in Table 1. The dialogue act groups
that we use are a subset of those used in [7]. We filtered out utterances in
the dataset that do not have corresponding audio segments (i.e., no timing
information). We obtain the final utterances by trimming the audio of the
appropriate speaker channel in the original media files in accordance to the
timings provided in the NXT Switchboard corpus.

Analysis. The final dataset that we use contains a total of 86,687 utterances.
Table 2 shows a summary of the content of the dataset in terms of turn labels
and speaker intentions. As a first pass to understand the relationship between
dialogue acts and turns, we run a Chi-square test of independence and find
that there is a relationship between dialogue acts and turns, $p<0.001$ (i.e.,
they are not independent). Note that this finding is suggestive rather than
conclusive; mainly because our utterances are not independent (they can come
from same speaker). Nevertheless, this finding supports findings in literature
[3, 12], which suggested that speaker intentions influence turn-taking
behavior.

Table 1: Mapping dialogue act classes to intention classes. SwDA classes | Intention classes
---|---
sd, h, bf | statement
sv, ad, sv@ | opinion
aa | agree
%, %- | abandon
b, bh | backchannel
qy, qo, qh | question
no, ny, ng, arp | answer
Table 2: Total number of utterances that are followed by holds and switches for each speaker intention class. Intention |  Counts ($\%$)
---|---
Holds | Switches
statement | 26,332 (52.2) | 12,722 (35.1)
opinion | 8,066 (16.0) | 5,227 (14.4)
agree | 3,997 (7.9) | 1,417 (3.9)
abandon | 3,887 (7.7) | 3,203 (8.8)
backchannel | 6,225 (12.3) | 10,678 (29.5)
question | 752 (1.5) | 2,369 (6.5)
answer | 1,197 (2.4) | 615 (1.7)
total | 50,456 | 36,231

###  4.2 Features

We use the OpenSMILE toolkit [15] to extract the following features and their
first (left) derivatives using a $25$ms Hamming window with a shift-rate of
$10$ms: intensity, loudness, MFCC, RMS energy, zero-crossing-rate, and
smoothed pitch. As a result, a given signal is represented as a sequence of
$42$-dimensional feature vectors. The choice of these features was inspired by
their success in previous studies on modeling turn-taking using acoustic cues
[4, 11].

##  5 Method

We use unidirectional LSTM network to model the sequence of acoustic features
and make turn predictions. LSTMs are able to capture past signal behavior and
they have shown success in many audio processing applications, such as speech
recognition and computational paralinguistics [6, 16]. In addition to their
ability to capture past signal behavior, LSTMs are able to capture information
relating to timing and differentials (e.g., rising slope); both of which are
useful for modeling turn-taking [3].

Predicting turns can be formulated as a binary classification task where the
goal is, given an utterance, predict whether there will be a turn-switch or a
turn-hold. We augment this task given the following problem setup: given an
utterance, simultaneously predict turn-transitions and speaker intentions. The
model is trained to minimize a joint loss function that takes the following
form:

$L_{tot}=\lambda_{1}L_{turn}+\lambda_{2}L_{intent}$

where $L_{turn}$ is the loss function for turn predictions, $L_{intent}$ is
the loss function for speaker intention predictions, $L_{tot}$ is the overall
loss function, $\lambda_{1}$ and $\lambda_{2}$ are weights assigned to control
the influence of each loss function. In this work we set $\lambda_{1}$ to
$1.0$ and $\lambda_{2}$ to $0.5$.

Baselines. We re-implement the “full model” from [7] and compare its
performance to the proposed approach. The full model uses a Random Forest
classifier with the features described in Section 2. We also compare our
proposed multi-task approach to a single-task LSTM that is trained to minimize
$L_{turn}$ alone. We note that the single-task LSTM approach is similar to the
one used in [4].

##  6 Experiments

###  6.1 Setup

We evaluate performance using 5-fold cross-validation. We split on
conversations, as opposed to utterances, to ensure that individual speakers do
not appear in both the training and testing folds. For each testing fold, we
randomly take out $33\%$ of the training conversations and use them for
validation and early stopping. For each conversation, we perform speaker-
specific $z$-normalization on the features.

We implement our models using the PyTorch
library222https://github.com/pytorch/pytorch. We optimize the weighted
negative log-likelihood loss function and use RMSProp optimizer to train our
models. We use an initial learning rate of $0.001$. At the end of each epoch,
we compute the macro-F1 score on the validation set and reduce the learning
rate by a factor of $2$ if there was no improvement from last epoch. We run
for a maximum of $100$ epochs and stop training if there was no improvement in
validation F1 score for $5$ consecutive epochs. We take a snapshot of the
model after each epoch and select the one that gave the highest validation
performance.

For each fold, we perform a grid search and pick the hyper-parameters that
maximize validation performance. The main hyper-parameters of the model are:
number of layers $\\{1,2\\}$ and layer width $\\{32,64,128\\}$. Once we have
identified the optimal hyper-parameters for each fold, we train $3$ models
with different random seeds and report their ensemble performance to minimize
variance due to random initialization. We report the average performance
across the five folds.

###  6.2 Results and Discussion

Table 3: Performance comparison of different methods. Results shown are macro-
averages across turn-switches and turn-holds.

Method | Rec. | Prec. | F1 | AUC
---|---|---|---|---
Random | $50.0$ | $49.6$ | $41.7$ | $45.3$
Full model [7] | $55.8$ | $56.7$ | $55.4$ | $57.8$
LSTM | $65.9$ | $65.6$ | $65.5$ | $71.9$
MT-LSTM | $\textbf{66.4}^{*}$ | 66.0 | 65.8 | $\textbf{72.6}^{*}$

∗ indicates $p<0.05$ under a paired $t$-test with LSTM.

Table 3 shows the results obtained from our experiments. The table shows that
a single-task LSTM, which uses the input features described in Section 4.2,
outperforms the full model in all evaluation metrics. We attribute this
improvements to better feature representations and better sequential modeling
abilities of LSTMs. The table shows that a multi-task LSTM, which is trained
using a joint loss function, provides consistent improvements over a single-
task LSTM (significant improvements under a paired $t$-test, $p<0.05$, in
terms of recall and AUC). This suggests that a turn prediction model can
benefit from representations extracted for detecting speaker intentions. Next,
we study how well our model is able to identify turn-switches when the
switches are smooth and when they are overlapping. Our model identifies turn-
switches with a recall of $68.5\%$ when the switches are overlapping and
identifies turn-switches with a recall of $68.1\%$ when the switches are
smooth.

Table 4 shows the performance of predicting turn-switches and turn-holds for
each intention class, as well as the accuracy of detecting that intention
class. The results show that the model is better able to predict turn-switches
when presented with a backchannel or a question, and is better able to predict
turn-holds when presented with a statement, opinion, or an answer. This
suggests that the performance of the model depends on the context and nature
of a dialogue, and that it is easier to anticipate turn-switches or turn-holds
for some intentions and not for others.

Table 4 also shows the performance of identifying speaker intentions by the
auxiliary task. The table shows that it is easier to identify backchannels,
questions, or turn-exit signals (abandon) than it is to identify agreement
signals and statements. The auxiliary task obtains an unweighted average
recall (UAR) of $45.6\%$ on a $7$-way classification task (where chance UAR is
$14.3\%$).

Table 4: Detecting turn-switches and turn-holds for each speaker intention class, as well as the per-class accuracy for detecting intentions by the auxiliary task. | Turn-Transitions | Intentions
---|---|---
F1 (switch) | F1 (hold) | per-class Acc.
statement | $51.4$ | $73.2$ | $39.6$
opinion | $54.5$ | $70.3$ | $43.1$
agree | $49.7$ | $66.5$ | $30.4$
abandon | $67.2$ | $68.3$ | $56.8$
backchannel | $79.1$ | $51.6$ | $49.6$
question | $72.2$ | $47.6$ | $50.3$
answer | $61.1$ | $71.7$ | $43.9$

##  7 Conclusion

In this work we showed that a model that uses acoustic features for modeling
turn-taking in spoken dialogues could benefit from adding speaker intention
detection as an auxiliary task. We also explored how the performance of our
turn-taking model varies depending on speaker intentions. For future work, we
plan to augment acoustic features with lexical or phonetic information. We
also plan to investigate combining turn-taking with end-of-utterance
detection. Finally, we plan to add our model to a live spoken dialogue system.

Acknowledgment. This work was supported by IBM under the Sapphire project.

## References

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  * [2] Simon Garrod and Martin J Pickering,  “The use of content and timing to predict turn transitions,”  Frontiers in psychology, vol. 6, pp. 751, 2015.
  * [3] Agustín Gravano and Julia Hirschberg,  “Turn-taking cues in task-oriented dialogue,”  Computer Speech & Language, vol. 25, no. 3, pp. 601–634, 2011\.
  * [4] Angelika Maier, Julian Hough, and David Schlangen,  “Towards deep end-of-turn prediction for situated spoken dialogue systems,”  Interspeech, 2017.
  * [5] Chaoran Liu, Carlos Ishi, and Hiroshi Ishiguro,  “Turn-taking estimation model based on joint embedding of lexical and prosodic contents,”  Interspeech, pp. 1686–1690, 2017.
  * [6] Carolina Parada, Gabor Simko, Matt Shannon, and Shuo-yiin Chang,  “Improved end-of-query detection for streaming speech recognition,”  pp. 2900–2904, 2017.
  * [7] Tomer Meshorer and Peter A Heeman,  “Using past speaker behavior to better predict turn transitions.,”  in Interspeech, 2016, pp. 2900–2904.
  * [8] Nishitha Guntakandla and Rodney Nielsen,  “Modelling turn-taking in human conversations,”  in AAAI Spring Symposium on Turn-Taking and Coordination in Human-Machine Interaction, Stanford CA, 2015.
  * [9] Ryo Masumura, Taichi Asami, Hirokazu Masataki, Ryo Ishii, and Ryuichiro Higashinaka,  “Online end-of-turn detection from speech based on stacked time-asynchronous sequential networks,”  Interspeech, pp. 1661–1665, 2017.
  * [10] Yuichi Ishimoto, Takehiro Teraoka, and Mika Enomoto,  “End-of-utterance prediction by prosodic features and phrase-dependency structure in spontaneous japanese speech,”  Interspeech, pp. 1681–1685, 2017.
  * [11] Harish Arsikere, Elizabeth Shriberg, and Umut Ozertem,  “Enhanced end-of-turn detection for speech to a personal assistant,”  in AAAI Spring Symposium on Turn-taking and Coordination in Human-Machine Interaction, 2015.
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#
Pseudospin and Spin Symmetric Solutions of Dirac Equation: Hellmann Potential,
Wei Hua Potential, Varshni Potential

Altuð Arda
<EMAIL_ADDRESS>
Ramazan Sever
<EMAIL_ADDRESS>

Approximate analytical solutions of the Dirac equation are obtained for the
Hellmann potential, Wei Hua potential and Varshni potential with any
$\kappa$-value for the cases having the Dirac equation pseudospin and spin
symmetries. Closed forms of the energy eigenvalue equations and the spinor
wave functions are obtained by using the Nikiforov-Uvarov method and some
tables are given to see the dependence of the energy eigenvalues on different
quantum number pairs ($n,\kappa$).

Pseudospin and Spin Symmetric Solutions of Dirac Equation: Hellmann Potential,
Wei Hua Potential, Varshni Potential

Altuğ Arda
<EMAIL_ADDRESS>
Ramazan Sever
<EMAIL_ADDRESS>

##  1 Introduction

The pseudospin and spin symmetric solutions of the Dirac equation have been
great interest in literature for last decades [1, 2, 3]. The Dirac equation
with vector, $V(r)$, and scalar, $S(r)$, potentials has pseudospin (spin)
symmetry when the difference (the sum) of the potentials $V(r)-S(r)$
$[V(r)+S(r)]$ is constant, which means $\frac{d}{dr}[V(r)-S(r)]=0$ (or
$\frac{d}{dr}[V(r)+S(r)]=0$). It is pointed out that these symmetries can
explain degeneracies in single-particle energy levels in nuclei or in some
heavy meson-spectra within the contexts of relativistic mean-field theories
[1, 2, 3]. In the relativistic domain, these symmetries were used in the
context of deformation and superdeformation in nuclei, magnetic moment
interpretation and identical bands [4]. In the non-relativistic domain,
performing a helicity unitary transformation to a single-particle Hamiltonian
maps the normal state onto the pseudo-state [5]. Moreover, the Dirac
Hamiltonian has not only a spin symmetry but also a $U(3)$ symmetry for the
case $V(r)=S(r)$ while not only a pseudospin symmetry but also a pseudo-$U(3)$
symmetry with vector and scalar harmonic oscillator potentials [6, 7]. Because
of these investigations, the solutions of the Dirac equation having spin and
pseudospin symmetry have received great attention for different type of
potentials such as Morse, Eckart, the modified Pöschl-Teller, the Manning-
Rosen potentials and the symmetrical well potential [8, 9, 10, 11, 12, 13, 14,
15].

Throughout the paper we use the following approximation instead of the spin-
orbit coupling term to obtain the analytical solutions of the Hellmann
potential, Wei Hua and Varshni potentials [16, 17, 18, 19, 20, 21, 22, 23]

$\displaystyle\frac{1}{r^{2}}\approx\beta^{2}\,\frac{1}{(1-e^{-\beta
r})^{2}}\,,$ (1)

where $\beta$ is a parameter related with the above potentials.

The potentials studied in the present work and also other some exponential-
type potentials such as a ring-shaped Hùlthen, the Yukawa and Tietz-Hua
potentials have been analyzed in details by using different methods [24, 25,
26, 27, 28, 29, 30]. We intend to use the Nikiforov-Uvarov method (NU) to
analyze the bound states of the Dirac equation for the cases of pseudospin and
spin symmetries. This method is a powerful tool to solve a second-order
differential equation and has been used to find the bound states of different
potentials in literature [31, 32].

The organization of this work is as follows. In Section 2, we briefly give the
Dirac equation with attractive scalar and repulsive vector potentials for the
cases where the Dirac equation has pseudospin and spin symmetries,
respectively. In Section 3, we present the NU method and the parameters
required within the method. In Section 4, we find an analytical energy
eigenvalue equations for the bound states and the two-component spinor wave
functions of the above potentials by using an approximation instead of the
spin-orbit coupling term. In Section 5, we give our results and discussions.
The last section includes our conclusions.

##  2 Dirac Equation

The free particle Dirac equation is given by ($\hbar=c=1$)

$\displaystyle\bigl{(}i\gamma^{\mu}\partial_{\mu}-M\bigr{)}\Psi(\vec{r},t)=0\,,$
(2)

Taking the total wave function as $\Psi(\vec{r},t)=e^{-iEt}\psi(\vec{r})$ for
time-independent potentials, where $E$ is the relativistic energy, $M$ is
particle mass, the Dirac equation with spherical symmetric vector and scalar
potentials is written as

$\displaystyle\bigl{[}\vec{\alpha}.\vec{P}+\beta(M+S(r))\bigr{]}\psi(\vec{r})=\big{[}E-V(r)\bigr{]}\psi(\vec{r})\,,$
(3)

Here $\alpha$ and $\beta$ are usual $4\times 4$ matrices. For spherical
nuclei, the angular momentum $\vec{J}$ and the operator
$\hat{K}=-\beta\bigl{(}\hat{\sigma}.\hat{L}+1\bigr{)}$ with eigenvalues
$\kappa=\pm(j+1/2)$ commute with the Dirac Hamiltonian, where $\hat{L}$ is the
orbital angular momentum. By using the radial eigenfunctions for upper and
lower components of the Dirac eigenfunction $F(r)$ and $G(r)$, respectively,
the wave function is written as [31]

$\displaystyle\psi(\vec{r})=\,\frac{1}{r}\,\Bigg{[}\begin{array}[]{c}\,F\,(r)Y^{(1)}(\theta,\phi)\\\
iG\,(r)Y^{(2)}(\theta,\phi)\end{array}\Bigg{]}\,,$ (6)

where $Y^{(1)}(\theta,\phi)$ and $Y^{(2)}(\theta,\phi)$ are the pseudospin and
spin spherical harmonics, respectively. They correspond to angular and spin
parts of the wave function given by

$\displaystyle Y^{(1),(2)}(\theta,\phi)=\sum_{m_{\ell}m_{s}}<\ell
m_{\ell}\frac{1}{2}m_{s}|\ell\frac{1}{2}jm>Y_{\ell
m_{\ell}}(\theta,\phi)\chi_{\frac{1}{2}m_{s}}\,,$ $\displaystyle
j=|\kappa|-\frac{1}{2}\,,\,\,\,\ell=\kappa\,\,(\kappa>0)\,;\,\ell=-(\kappa+1)\,\,(\kappa<0)\,,$
(7)

Here, $Y_{\ell m_{\ell}}(\theta,\phi)$ denotes the spherical harmonics and
$m_{\ell}$ and $m_{s}$ are related magnetic quantum numbers.

Substituting Eq. (4) into Eq. (3) gives us the following coupled differential
equations

$\displaystyle\left(\frac{d}{dr}+\frac{\kappa}{r}\right)F(r)=[E+M-\Gamma(r)]G(r)\,,$
(8a)
$\displaystyle\left(\frac{d}{dr}-\frac{\kappa}{r}\right)G(r)=[M-E+\Lambda(r)]F(r)\,.$
(8b)

where $\Gamma(r)=V(r)-S(r)$ and $\Lambda(r)=V(r)+S(r)$. Using the expression
$G(r)$ in Eq. (6a) and inserting it into Eq. (6b), we get a second order
differential equation

$\displaystyle\left[\frac{d^{2}}{dr^{2}}-\frac{\kappa(\kappa+1)}{r^{2}}+\varepsilon^{(1)}(r)\right]F(r)=-\left[\frac{d\Gamma(r)/dr}{\left[E+M-\Gamma(r)\right]}\right]F(r)\,,$
(9)

where
$\varepsilon^{(1)}(r)=\left[E+M-\Gamma(r)\right]\left[E-M-\Lambda(r)\right]$.
By similar steps, we write the following second order differential equation
for $G(r)$ as

$\displaystyle\left[\frac{d^{2}}{dr^{2}}-\frac{\kappa(\kappa-1)}{r^{2}}+\varepsilon^{(2)}(r)\right]G(r)=\left[\frac{d\Lambda(r)/dr}{\left[M-E+\Lambda(r)\right]}\right]G(r)\,,$
(10)

where
$\varepsilon^{(2)}(r)=\left[E-M-\Lambda(r)\right]\left[E+M-\Gamma(r)\right]$.
If the Dirac equation has spin symmetry which means that $\Gamma(r)=A_{1}$
($d\Gamma(r)/dr=0$) is a constant, Eq. (7) has the following form

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\kappa(\kappa+1)}{r^{2}}+\left[E+M-A_{1}\right]\left[E-M-\Lambda(r)\right]\right\\}F(r)=0\,,$
(11)

and if the Dirac equation has pseudospin symmetry which means that
$\Lambda(r)=A_{2}$ ($d\Lambda(r)/dr=0$) is a constant, Eq. (8) becomes

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\kappa(\kappa-1)}{r^{2}}+\left[E-M-
A_{2}\right]\left[E+M-\Gamma(r)\right]\right\\}G(r)=0\,.$ (12)

##  3 Nikiforov Uvarov Method

The Nikiforov-Uvarov method could be used to solve a second-order differential
equation of the hypergeometric-type which can be transformed by using
appropriate coordinate transformation into the following form

$\displaystyle\sigma^{2}(z)\frac{d^{2}\Psi(z)}{dz^{2}}+\sigma(z)\tilde{\tau}(z)\frac{d\Psi(z)}{dz}+\tilde{\sigma}(z)\Psi(z)=0\,,$
(13)

where $\sigma(z)$ , and $\tilde{\sigma}(z)$ are polynomials, at most, second
degree, and $\tilde{\tau}(z)$ is a first-degree polynomial. By taking the
solution as

$\displaystyle\Psi(z)=\psi(z)\varphi(z)\,,$ (14)

gives Eq. (11) as a hypergeometric type equation [32]

$\displaystyle\frac{d^{2}\varphi(z)}{dz^{2}}+\frac{\tau(z)}{\sigma(z)}\frac{d\varphi(z)}{dz}+\frac{\lambda}{\sigma(z)}\,\varphi(z)=0\,,$
(15)

where $\psi(z)$ is defined by using the equation [32]

$\displaystyle\frac{1}{\psi(z)}\frac{d\psi(z)}{dz}=\frac{\pi(z)}{\sigma(z)}\,,$
(16)

and the other part of the solution in Eq. (12) is given by

$\displaystyle\varphi_{n}(z)=\frac{a_{n}}{\rho(z)}\frac{d^{n}}{dz^{n}}[\sigma^{n}(z)\rho(z)]\,,$
(17)

where $a_{n}$ is a normalization constant, and $\rho(z)$ is the weight
function, and satisfies the following equation [32]

$\displaystyle\frac{d\sigma(z)}{dz}+\frac{\sigma(z)}{\rho(z)}\frac{d\rho(z)}{dz}=\tau(z)\,.$
(18)

The function $\pi(z)$ and the parameter $\lambda$ in the above equation are
defined as

$\displaystyle\pi(z)$ $\displaystyle=$
$\displaystyle\,\frac{1}{2}\,[\frac{d}{dz}\,\sigma(z)-\tilde{\tau}(z)]\pm\bigg{\\{}\frac{1}{4}\left[\frac{d}{dz}\,\sigma(z)-\tilde{\tau}(z)\right]^{2}-\tilde{\sigma}(z)+k\sigma(z)\bigg{\\}}^{1/2}\,,$
(19) $\displaystyle\lambda$ $\displaystyle=$ $\displaystyle
k+\frac{d}{dz}\,\pi(z)\,.$ (20)

In the NU method, the square root in Eq. (17) must be the square of a
polynomial, so the parameter $k$ can be determined. Thus, a new eigenvalue
equation becomes

$\displaystyle\lambda=\lambda_{n}=-n\frac{d}{dz}\,\tau(z)-\frac{1}{2}\,(n^{2}-n)\frac{d^{2}}{dz^{2}}\,\sigma(z)\,.$
(21)

and the derivative of the function $\tau(z)=\tilde{\tau}(z)+2\pi(z)$ should be
negative.

##  4 Bound State Solutions

###  4.1 Hellmann Potential

The Hellmann potential having the form

$\displaystyle V(r)=-\frac{a}{r}+\frac{b}{r}\,e^{-\beta r}\,,$ (22)

has been used to explain the electron-ion [33] or electron-core interaction
[34], alkali hydride molecules and to study of inner-shell ionisation problem
[35]. We present the plot of the above potential in Fig. (1) to see the
variation with position coordinate.

1\. Spin Symmetric Solutions

Inserting Eq. (20) into Eq. (9) and using the approximation given in Eq. (1)
instead of the spin-orbit coupling term, we obtain

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\beta^{2}\kappa(\kappa+1)}{(1-e^{-\beta
r})^{2}}+\frac{\beta}{1-e^{-\beta r}}\left(a-be^{-\beta
r}\right)+\epsilon_{H}^{SS}\right\\}F(r)=0\,,$ (23)

where $H$ stands for the Hellmann potential and
$\epsilon_{H}^{SS}=(E+M-A_{1})(E-M)$. Defining a new variable $z=e^{-\beta r}$
and using the following abbreviations

$\displaystyle
a^{2}_{1}=\kappa(\kappa+1)-\frac{1}{\beta^{2}}\left(a\beta+\epsilon_{H}^{SS}\right)\,,$
(24a) $\displaystyle
a^{2}_{2}=\frac{1}{\beta^{2}}\left[\beta(a+b)+2\epsilon_{H}^{SS}\right]\,,$
(24b) $\displaystyle
a^{2}_{3}=-\frac{1}{\beta^{2}}\left[b\beta+\epsilon_{H}^{SS}\right]\,,$ (24c)

we write Eq. (21) as

$\displaystyle\frac{d^{2}F(z)}{dz^{2}}+\frac{1-z}{z(1-z)}\frac{dF(z)}{dz}+\frac{1}{z^{2}(1-z)^{2}}\left[-a^{2}_{1}-a^{2}_{2}z-a^{2}_{3}z^{2}\right]F(z)=0\,,$
(25)

Comparing the last equation with Eq. (11), we have

$\displaystyle\tilde{\tau}(z)=1-z\,,\,\,\,\,\,\sigma(z)=z(1-z)\,,\,\,\,\,\,\tilde{\sigma}(z)=-a_{1}^{2}z^{2}-a_{2}^{2}z-a_{3}^{2}\,,$
(26)

The function $\pi(z)$ is obtained from Eq. (17) as

$\displaystyle\pi(z)=\,-\frac{1}{2}\,z\,\mp\sqrt{(\,\frac{1}{4}\,+a_{3}^{2}-k)z^{2}+(a_{2}^{2}+k)z+a_{1}^{2}\,}\,,$
(27)

The constant $k$ is determined by imposing a condition such that the
discriminant under the square root should be zero. The roots of $k$ are
$k_{1,2}=-a_{2}^{2}-2a_{1}^{2}\mp a_{1}(1+2\kappa)$. Substituting the value of
$k_{1}=-a_{2}^{2}-2a_{1}^{2}+a_{1}(1+2\kappa)$ into Eq. (25), we get for
$\pi(z)$

$\displaystyle\pi(z)(k\rightarrow
k_{1})=\left\\{\begin{array}[]{lr}-(a_{1}-\kappa)z+a_{1}&\\\ -(1+\kappa-
a_{1})z-a_{1}\,,&\\\ \end{array}\right.$ (30)

Now we calculate the polynomial $\tau(z)$ from $\pi(z)$ such that its
derivative with respect to $z$ must be negative. Thus we obtain $\tau(z)$ for
the second choice in last equation as

$\displaystyle\tau(z)=(2a_{1}-1)z-(1+2\kappa+2a_{1})\,,$ (31)

The constant $\lambda$ in Eq. (18) becomes

$\displaystyle\lambda=-a^{2}_{2}-2a_{1}^{2}+a_{1}(1+2\kappa)-a_{1}+\kappa\,,$
(32)

and Eq. (19) gives us

$\displaystyle\lambda_{n}=n(n-2a_{1})\,.$ (33)

Substituting the values of the parameters given by Eq. (22), and setting
$\lambda=\lambda_{n}$, one can find the energy eigenvalues for the Hellmann
potential as

$\displaystyle
E=\frac{1}{2}\left[A_{1}\mp\sqrt{A_{1}^{2}-4(MA_{1}-M^{2}-N)\,}\right]\,,$
(34)

where $N$ is a parameter written in terms of the quantum numbers $n$ and
$\kappa$ as

$\displaystyle
N=-\frac{\beta^{2}}{4(n+\kappa)^{2}}\left[\frac{1}{\beta}(a-b)-(n^{2}-\kappa^{2})-\kappa(\kappa+1)\right]^{2}-a\beta+\beta^{2}\kappa(\kappa+1)\,.$
(35)

Now we find the upper component of the Dirac wave function. We first compute
the weight function from Eq. (16) with the help of Eq. (27)

$\displaystyle\rho(z)=z^{-2(1+\kappa+a_{1})}\,(1-z)^{(1+2\kappa)}\,,$ (36)

and we obtain from Eq. (15)

$\displaystyle\varphi_{n}(z)\sim
z^{-2(1+\kappa+a_{1})}\,(1-z)^{(1+2\kappa)}\,\frac{d^{n}}{dz^{n}}\,\left[\,z^{n-\kappa-
a_{1}-2}\,(1-z)^{n-2\kappa-1}\right]\,,$ (37)

The polynomial solutions can be written in terms of the Jacobi polynomials
[38]

$\displaystyle\varphi_{n}(z)\sim
P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,.$ (38)

The other part of the wave function is obtained from the Eq. (15) as

$\displaystyle\psi(z)=z^{-a_{1}}(1-z)^{1-\kappa}\,,$ (39)

Thus we write the upper component for the Hellmann potential in Eq. (4) as

$\displaystyle F(z)\sim
z^{-a_{1}}(1-z)^{1-\kappa}P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,,$
(40)

By using Eq. (6a) and the identity for derivative of the Jacobi polynomials
given as
$\frac{d}{dx}P_{n}^{(p,q)(x)}=\frac{1}{2}(n+p+q+1)P_{n-1}^{(p+1,q+1)}(x)$
[38], we obtain the other component for the Hellmann potential as

$\displaystyle G(z)$
$\displaystyle\sim\frac{z^{-a_{1}}(1-z)^{1-\kappa}}{E+M-A}[\beta(\frac{1}{a_{1}}-\frac{\kappa}{lnz})P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)$
(41)
$\displaystyle-\frac{1}{4}(n-2a_{1})P_{n}^{(-(1+2\kappa+2a_{1}),\,-(2+2\kappa)\,)}\,(1-2z)]\,.$

2\. Pseudospin Symmetric Solutions

Inserting Eq. (20) into Eq. (10) and using the approximation given in Eq. (1),
we obtain

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\beta^{2}\kappa(\kappa-1)}{(1-e^{-\beta
r})^{2}}+\frac{\beta}{1-e^{-\beta r}}\left(a-be^{-\beta
r}\right)+\epsilon_{H}^{PSS}\right\\}F(r)=0\,,$ (42)

where $\epsilon_{H}^{PSS}=(E-M-A_{2})(E+M)$. Using the same variable and the
following abbreviations

$\displaystyle
a^{2}_{1}=\kappa(\kappa-1)-\frac{1}{\beta^{2}}\left(a\beta+\epsilon_{H}^{PSS}\right)\,,$
(43a) $\displaystyle
a^{2}_{2}=\frac{1}{\beta^{2}}\left[\beta(a+b)+2\epsilon_{H}^{PSS}\right]\,,$
(43b) $\displaystyle
a^{2}_{3}=-\frac{1}{\beta^{2}}\left[b\beta+\epsilon_{H}^{PSS}\right]\,,$ (43c)

we obtain

$\displaystyle\frac{d^{2}G(z)}{dz^{2}}+\frac{1-z}{z(1-z)}\frac{dG(z)}{dz}+\frac{1}{z^{2}(1-z)^{2}}\left[-a^{2}_{1}-a^{2}_{2}z-a^{2}_{3}z^{2}\right]G(z)=0\,,$
(44)

Following the same steps in previous section, we write the energy eigenvalues
for the Hellmann potential for the case of pseudospin symmetry

$\displaystyle
E=\frac{1}{2}\left[A_{2}\mp\sqrt{A_{2}^{2}+4(MA_{2}+M^{2}+N)\,}\right]\,,$
(45)

where $N$ is given as

$\displaystyle
N=-\frac{\beta^{2}}{4(n+\kappa)^{2}}\left[-\frac{1}{\beta}(a-b)+n^{2}+\kappa^{2}+\kappa(\kappa-3)\right]^{2}-a\beta+\beta^{2}\kappa(\kappa-1)\,.$
(46)

and the lower component is written as

$\displaystyle G(z)\sim
z^{-a_{1}}(1-z)^{1-\kappa}P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,.$
(47)

Using Eq. (6b) gives us the other component as

$\displaystyle F(z)$
$\displaystyle\sim\frac{z^{-a_{1}}(1-z)^{1-\kappa}}{M-E+A}[\beta(\frac{1}{a_{1}}-\frac{\kappa}{lnz})P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)$
(48)
$\displaystyle-\frac{1}{4}(n-2a_{1})P_{n}^{(-(1+2\kappa+2a_{1}),\,-(2+2\kappa)\,)}\,(1-2z)]\,.$

###  4.2 Wei Hua Potential

The Wei Hua potential is written

$\displaystyle V(r)=D\left[\frac{1-e^{-\beta r}}{1-ae^{-\beta
r}}\right]^{2}\,,$ (49)

which is proposed for bond-stretching vibration of diatomic molecules [36]. We
give the plot of the Wei Hua potential in Fig. (2).

1\. Spin Symmetric Solutions

Inserting last equation and Eq. (1) into Eq. (9), we obtain

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\beta^{2}\kappa(\kappa+1)}{(1-e^{-\beta
r})^{2}}-D\left[\frac{1-e^{-\beta r}}{1-ae^{-\beta
r}}\right]^{2}+\epsilon_{WH}^{SS}\right\\}F(r)=0\,,$ (50)

where $WH$ stands for the Wei Hua potential and
$\epsilon_{WH}^{SS}=(E+M-A_{1})(E-M)$. Defining a new variable $z=ae^{-\beta
r}$, using the abbreviations

$\displaystyle
a^{2}_{1}=\kappa(\kappa+1)-\frac{1}{\beta^{2}}\left(\epsilon_{WH}^{SS}-D\right)\,,$
(51a) $\displaystyle
a^{2}_{2}=-\frac{1}{\beta^{2}}\left[\frac{2D}{a}-2\epsilon_{WH}^{SS}\right]\,,$
(51b) $\displaystyle
a^{2}_{3}=-\frac{1}{\beta^{2}}\left[\epsilon_{WH}^{SS}-\frac{D}{a^{2}}\right]\,,$
(51c)

and following the same procedure in the above section for the Hellmann
potential, we write the energy eigenvalues of the Wei Hua potential for the
case of spin symmetry

$\displaystyle
E=\frac{1}{2}\left[A_{1}\mp\sqrt{A_{1}^{2}-4(MA_{1}-M^{2}-N)\,}\right]\,,$
(52)

where $N$ is a parameter written in terms of the quantum numbers $n$ and
$\kappa$ as

$\displaystyle
N=-\frac{\beta^{2}}{4(n+\kappa)^{2}}\left[n^{2}+\kappa^{2}+\kappa(\kappa+1)-\frac{2D}{\beta^{2}}(\frac{1}{a}-1)\right]^{2}+D+\beta^{2}\kappa(\kappa+1)\,.$
(53)

and the lower component for the Wei Hua potential

$\displaystyle F(z)\sim
z^{-a_{1}}(1-z)^{1-\kappa}P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,.$
(54)

By using Eq. (6a) we obtain the other component as

$\displaystyle G(z)$
$\displaystyle\sim\frac{z^{-a_{1}}(1-z)^{1-\kappa}}{E+M-A}[\beta(\frac{1}{a_{1}}-\frac{\kappa}{lnz})P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)$
(55)
$\displaystyle-\frac{1}{4}(n-2a_{1})P_{n}^{(-(1+2\kappa+2a_{1}),\,-(2+2\kappa)\,)}\,(1-2z)]\,.$

2\. Pseudospin Symmetric Solutions

Inserting Eqs. (45) and (1) into Eq. (10), we obtain

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\beta^{2}\kappa(\kappa-1)}{(1-e^{-\beta
r})^{2}}-D\left[\frac{1-e^{-\beta r}}{1-ae^{-\beta
r}}\right]^{2}+\epsilon_{WH}^{PSS}\right\\}F(r)=0\,,$ (56)

$\epsilon_{WH}^{PSS}=(E-M-A_{2})(E+M)$. Using the same variable $z$ for the
Hellmann potential and defining the abbreviations

$\displaystyle
a^{2}_{1}=\kappa(\kappa-1)-\frac{1}{\beta^{2}}\left(\epsilon_{WH}^{PSS}-D\right)\,,$
(57a) $\displaystyle
a^{2}_{2}=-\frac{1}{\beta^{2}}\left[\frac{2D}{a}-2\epsilon_{WH}^{PSS}\right]\,,$
(57b) $\displaystyle
a^{2}_{3}=-\frac{1}{\beta^{2}}\left[\epsilon_{WH}^{PSS}-\frac{D}{a^{2}}\right]\,,$
(57c)

and following the same procedure in the above section for the Hellmann
potential, we write the energy eigenvalues of the Wei Hua potential for the
case of pseudospin symmetry

$\displaystyle
E=\frac{1}{2}\left[A_{2}\mp\sqrt{A_{2}^{2}+4(MA_{2}+M^{2}+N)\,}\right]\,,$
(58)

where $N$ is a parameter written in terms of the quantum numbers $n$ and
$\kappa$ as

$\displaystyle
N=-\frac{\beta^{2}}{4(n+\kappa)^{2}}\left[n^{2}-\kappa^{2}+\kappa(\kappa-1)-\frac{2D}{\beta^{2}}(\frac{1}{a}-1)\right]^{2}+D+\beta^{2}\kappa(\kappa-1)\,.$
(59)

and the upper component for the Wei Hua potential

$\displaystyle G(z)\sim
z^{-a_{1}}(1-z)^{1-\kappa}P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,.$
(60)

The other component can be obtained from Eq. (6b) as

$\displaystyle F(z)$
$\displaystyle\sim\frac{z^{-a_{1}}(1-z)^{1-\kappa}}{M-E+A}[\beta(\frac{1}{a_{1}}-\frac{\kappa}{lnz})P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)$
(61)
$\displaystyle-\frac{1}{4}(n-2a_{1})P_{n}^{(-(1+2\kappa+2a_{1}),\,-(2+2\kappa)\,)}\,(1-2z)]\,.$

###  4.3 Varshni Potential

Varshni, for the first time, proposed the following potential function

$\displaystyle V(r)=a\left[1-\frac{b}{r}\,e^{-\beta r}\right]\,,$ (62)

to study the diatomic molecules [37]. It is clearly seen that the potential is
very similar to the Hellmann potential which could be seen in Fig. (3). All
figures show that the form of the potentials presented in this work are very
similar.

Now we tend to study the spin and pseudospin symmetric solutions of the Dirac
equation for the above potential.

1\. Spin Symmetric Solutions

Inserting Eqs. (58) and Eq. (1) into Eq. (9), we obtain

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\beta}{1-e^{-\beta
r}}\left(\frac{\beta\kappa(\kappa+1)}{1-e^{-\beta r}}-abe^{-\beta
r}\right)+\epsilon_{V}^{SS}\right\\}F(r)=0\,,$ (63)

where $V$ stands for the Varshni potential and
$\epsilon_{V}^{SS}=(E+M-A_{1})(E-M)-a$. Defining a new variable $z=e^{-\beta
r}$, using the abbreviations

$\displaystyle
a^{2}_{1}=\kappa(\kappa+1)-\frac{\epsilon_{V}^{SS}}{\beta^{2}}\,,$ (64a)
$\displaystyle
a^{2}_{2}=-\frac{1}{\beta^{2}}\left[ab\beta-2\epsilon_{V}^{SS}\right]\,,$
(64b) $\displaystyle
a^{2}_{3}=-\frac{1}{\beta^{2}}\left[\epsilon_{V}^{SS}-ab\beta\right]\,,$ (64c)

and following the same procedure in the above sections, we write the energy
eigenvalues of the Varshni potential for the case of spin symmetry

$\displaystyle
E=\frac{1}{2}\left[A_{1}\mp\sqrt{A_{1}^{2}-4(MA_{1}-M^{2}-N)\,}\right]\,,$
(65)

where

$\displaystyle
N=-\frac{\beta^{2}}{4(n+\kappa)^{2}}\left[-\frac{ab}{\beta}+n^{2}+\kappa^{2}+\kappa(\kappa+1)\right]^{2}+\beta^{2}\kappa(\kappa+1)+a\,.$
(66)

and the lower component for the Varshni potential

$\displaystyle F(z)\sim
z^{-a_{1}}(1-z)^{1-\kappa}P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,.$
(67)

By using Eq. (6a) we obtain the other component as

$\displaystyle G(z)$
$\displaystyle\sim\frac{z^{-a_{1}}(1-z)^{1-\kappa}}{E+M-A}[\beta(\frac{1}{a_{1}}-\frac{\kappa}{lnz})P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)$
(68)
$\displaystyle-\frac{1}{4}(n-2a_{1})P_{n}^{(-(1+2\kappa+2a_{1}),\,-(2+2\kappa)\,)}\,(1-2z)]\,.$

2\. Pseudospin Symmetric Solutions

Inserting Eqs. (58) and (1) into Eq. (10), we obtain

$\displaystyle\left\\{\frac{d^{2}}{dr^{2}}-\frac{\beta}{1-e^{-\beta
r}}\left(\frac{\beta\kappa(\kappa-1)}{1-e^{-\beta r}}-abe^{-\beta
r}\right)+\epsilon_{V}^{PSS}\right\\}F(r)=0\,,$ (69)

$\epsilon_{V}^{PSS}=(E-M-A_{2})(E+M)$. Using the same variable $z$ for the
Hellmann potential and defining the abbreviations

$\displaystyle
a^{2}_{1}=\kappa(\kappa-1)-\frac{\epsilon_{V}^{PSS}}{\beta^{2}}\,,$ (70a)
$\displaystyle
a^{2}_{2}=-\frac{1}{\beta^{2}}\left[ab\beta-2\epsilon_{V}^{PSS}\right]\,,$
(70b) $\displaystyle
a^{2}_{3}=-\frac{1}{\beta^{2}}\left[\epsilon_{V}^{PSS}-ab\beta\right]\,,$
(70c)

and following the same procedure in the above sections, we write the energy
eigenvalues of the Varshni potential for the case of pseudospin symmetry

$\displaystyle
E=\frac{1}{2}\left[A_{2}\mp\sqrt{A_{2}^{2}+4(MA_{2}+M^{2}+N)\,}\right]\,,$
(71)

where

$\displaystyle
N=-\frac{\beta^{2}}{4(n+\kappa)^{2}}\left[n^{2}+\kappa^{2}+\kappa(\kappa-3)-\frac{ab}{\beta}\right]^{2}-\beta^{2}\kappa(\kappa-1)+a\,.$
(72)

and the upper component for the Varshni potential

$\displaystyle G(z)\sim
z^{-a_{1}}(1-z)^{1-\kappa}P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)\,.$
(73)

Using Eq. (6b) gives the other component as

$\displaystyle F(z)$
$\displaystyle\sim\frac{z^{-a_{1}}(1-z)^{1-\kappa}}{M-E+A}[\beta(\frac{1}{a_{1}}-\frac{\kappa}{lnz})P_{n}^{(-2(1+\kappa+a_{1}),\,-(1+2\kappa)\,)}\,(1-2z)$
(74)
$\displaystyle-\frac{1}{4}(n-2a_{1})P_{n}^{(-(1+2\kappa+2a_{1}),\,-(2+2\kappa)\,)}\,(1-2z)]\,.$

##  5 Results and Discussions

We have listed some numerical values for energy eigenvalues in Tables I-VI for
the cases of spin and pseudospin symmetries, separately. We have used the same
parameter values in both of spin and pseusospin symmetric cases for the
Hellmann potential, this is valid also for the Varshni potential. But the
values of the parameters for Wei Hua potential are different for the cases of
spin and pseudospin symmetries. It could be seen that the dependence of the
bound states for the Wei Hua potential are more sensitive. It also should be
stressed that the spin (and pseudospin) doublets, i.e., $(0,-2)$ and $(0,1)$
states or $(1,-2)$ and $(1,1)$ states, etc. could be seen up to fourth decimal
in energy eigenvalues.

##  6 Conclusion

We have studied the approximate bound state solutions of the Dirac equation
for the Hellmann potential, Wei Hua potential and Varshni potential, which
have an exponential form depending on the spatially coordinate $r$, for the
cases where the Dirac equation has pseudospin and spin symmetry, respectively.
The variation of the above potentials according to coordinate $r$ are given in
Figs I-III. We have obtained the energy eigenvalue equations and the related
two-component spinor wave functions with the help of Nikiforov-Uvarov method
and summarized the numerical results for the bound states in Tables I-VI. It
is also seen that the Nikiforov-Uvarov method is a suitable method to study
the bound state solutions of the above potentials.

##  7 Acknowledgments

This research was partially supported by the Scientific and Technical Research
Council of Turkey.

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Table 1: The energy eigenvalues of the Hellmann potential for the case of spin symmetry for $a=0.25,b=0.20,\beta=0.02,A_{1}=M=10$. $\ell$ | $n$ | $\kappa$ | $E>0$ | $n$ | $\kappa$ | $E>0$
---|---|---|---|---|---|---
1 | 0 | -2 | 9.9995294 | 0 | 1 | 9.9995575
2 |  | -3 | 9.9997604 |  | 2 | 9.9997394
3 |  | -4 | 9.9999536 |  | 3 | 9.9999700
4 |  | -5 | 10.0002770 |  | 4 | 10.0002900
1 | 1 | -2 | 9.9994575 | 1 | 1 | 9.9995700
2 |  | -3 | 9.9996894 |  | 2 | 9.9997300
3 |  | -4 | 9.9999464 |  | 3 | 9.9999700
4 |  | -5 | 10.0002740 |  | 4 | 10.0002900
Table 2:  The energy eigenvalues of the Hellmann potential for the case of pseudospin symmetry for $a=0.25,b=0.20,\beta=0.02,A_{2}=M=10$. $\ell$ | $n$ | $\kappa$ | $E>0$ | $n$ | $\kappa$ | $E>0$
---|---|---|---|---|---|---
1 | 0 | -2 | 9.9998031 | 0 | 1 | 9.9997925
2 |  | -3 | 9.9997710 |  | 2 | 9.9998598
3 |  | -4 | 9.9997412 |  | 3 | 9.9998977
4 |  | -5 | 9.9997125 |  | 4 | 9.9992950
1 | 1 | -2 | 9.9993925 | 1 | 1 | 9.9998281
2 |  | -3 | 9.9994514 |  | 2 | 9.9998599
3 |  | -4 | 9.9994310 |  | 3 | 9.9999016
4 |  | -5 | 9.9993933 |  | 4 | 9.9999477
Table 3:  The energy eigenvalues of the Wei Hua potential for the case of spin symmetry for $a=0.10,D=0.0001,\beta=0.01,A_{1}=2,M=0.001$. $\ell$ | $n$ | $\kappa$ | $E>0$ | $n$ | $\kappa$ | $E>0$
---|---|---|---|---|---|---
1 | 0 | -2 | 1.9986997 | 0 | 1 | 1.9963313
2 |  | -3 | 1.9993378 |  | 2 | 1.9991501
3 |  | -4 | 1.9995723 |  | 3 | 1.9996379
4 |  | -5 | 1.9996860 |  | 4 | 1.9997974
1 | 1 | -2 | 1.9976352 | 1 | 1 | 1.9985369
2 |  | -3 | 1.9993378 |  | 2 | 1.9996379
3 |  | -4 | 1.9994823 |  | 3 | 1.9996379
4 |  | -5 | 1.9994378 |  | 4 | 1.9998700
Table 4:  The energy eigenvalues of the Wei Hua potential for the case of pseudospin symmetry for $a=0.25,D=0.01,\beta=0.10,A_{2}=10,M=1$. $\ell$ | $n$ | $\kappa$ | $E>0$ | $n$ | $\kappa$ | $E>0$
---|---|---|---|---|---|---
1 | 0 | -2 | 1.0049979 | 0 | 1 | 0.9956234
2 |  | -3 | 1.0056224 |  | 2 | 1.0024995
3 |  | -4 | 1.0057785 |  | 3 | 1.0039570
4 |  | -5 | 1.0058222 |  | 4 | 1.0045295
1 | 1 | -2 | 1.0006250 | 1 | 1 | 1.0000000
2 |  | -3 | 0.9974994 |  | 2 | 1.0024763
3 |  | -4 | 0.9952527 |  | 3 | 1.0045295
4 |  | -5 | 0.9932780 |  | 4 | 1.0064216
Table 5:  The energy eigenvalues of the Varshni potential for the case of spin symmetry for $a=b=0.15,\beta=0.001,A_{1}=M=5$. $\ell$ | $n$ | $\kappa$ | $E>0$ | $n$ | $\kappa$ | $E>0$
---|---|---|---|---|---|---
1 | 0 | -2 | 4.9999970 | 0 | 1 | 4.9999814
2 |  | -3 | 5.0000009 |  | 2 | 4.9999992
3 |  | -4 | 5.0000023 |  | 3 | 5.0000024
4 |  | -5 | 5.0000030 |  | 4 | 5.0000034
1 | 1 | -2 | 4.9999884 | 1 | 1 | 4.9999961
2 |  | -3 | 5.0000070 |  | 2 | 5.0000050
3 |  | -4 | 5.0000022 |  | 3 | 5.0000024
4 |  | -5 | 5.0000023 |  | 4 | 5.0000036
Table 6:  The energy eigenvalues of the Varshni potential for the case of pseudospin symmetry for $a=b=0.15,\beta=0.001,A_{2}=M=5$. $\ell$ | $n$ | $\kappa$ | $E>0$ | $n$ | $\kappa$ | $E>0$
---|---|---|---|---|---|---
1 | 0 | -2 | 5.0000001 | 0 | 1 | 4.9999908
2 |  | -3 | 5.0000008 |  | 2 | 4.9999984
3 |  | -4 | 5.0000009 |  | 3 | 5.0000001
4 |  | -5 | 5.0000008 |  | 4 | 5.0000008
1 | 1 | -2 | 4.9999995 | 1 | 1 | 4.9999979
2 |  | -3 | 5.0000007 |  | 2 | 4.9999994
3 |  | -4 | 5.0000004 |  | 3 | 5.0000002
4 |  | -5 | 5.0000000 |  | 4 | 5.0000008

Figure 1: The Hellman potential. Figure 2: The Wei Hua potential. Figure 3:
The Varshni potential.