## Abstract

We studied a class of exponential potential model \(V(x)=-a\,e^{-b\,x}\) (\(a>0, b>0\)) and found that its solutions are given by the Bessel functions, but the energy spectra \(E=-b^2(n+1/2)^2/8\) which are derived from the quantization condition do not correspond to any discrete bound states. The energy levels which are calculated by the boundary condition \(J_{\nu }(2\sqrt{2a}/b)=0\) at the origin are in good agreement with the numerical results. We illustrate the wave functions through varying the potential parameters *a*, *b* and notice that they are pull back to the origin when the potential parameter *a* or *b* increases.

## Introduction

It is well known that the exact solutions of quantum systems can provide us useful information for understanding the fundamental concepts in quantum mechanics. The physicists often choose the hydrogen atom and one-dimensional harmonic oscillator to explain the basic quantum phenomena since the early foundation of quantum mechanics in 1920s [1, 2]. Recently, a lot of efforts have been given to soluble potentials via different approaches [3,4,5,6,7,8,9], which include the standard functional analysis method [1,2,3], SUSYQM approach [4], the factorization method [5], exact quantization rule method [6,7,8], proper quantization rule approach [9], the N-U method [10], asymptotic iteration method [11], the path integral method [12], the wave function Ansatz method [13, 14], group theory approach [15] and others.

Up to now, there are some important physical potential models involving the exponential type term such as the Morse potential [2], the Yukawa potential [16] (also named as the screened Coulomb potential), Deng-Fan potential [17], sine hyperbolic potentials [18], Rosen-Morse potential [19], improved multiparameter exponential type potential [20], Tietz potential [21], Wei Hua potential [22] and others [23, 24]. Nevertheless, the physical potential model of the exponential potential

does not consider an electromagnetic term 1/*x* (\(x>0\)) and is regarded as the simplest exponential potential model compared with those related to exponential type potentials mentioned above. This potential which is taken as a short-range interaction potential decays quickly with the increasing variable *x* [25]. As we know, the Yukawa potential is used widely in nuclear physics, while the other exponential type potentials have been applied to molecular physics, in particular diatomic molecules [2, 17,18,19,20,21,22]. The potential we are going to study is between the molecular and nuclear physics, which might be useful in atomic physics. The purpose of this work is to restudy this potential in order to enrich the study of this potential even though some studies were done in [3, 25] but we have to point out that the energy spectra \(E=-b^2(n+1/2)^2/8\) which are derived from the quantization condition do not correspond to any discrete bound states. This is very different from the traditional approach, i.e., the energy spectra can be obtained from the quantum quantization condition. For the present case, the energy spectra have to be calculated from the boundary condition \(J_{\nu }(2\sqrt{2a}/b)=0\) at the origin , which is not similar to our recent studies related to Heun differential equation [26,27,28,29,30].

The rest of this paper is organized as follows. In Sect. 2 we try to find the solutions of the Schrödinger equation with this potential. It is found that the solutions are given by the Bessel functions, but the energy spectra \(E=-b^2(n+1/2)^2/8\) which are derived from the quantization condition do not correspond to any discrete bound states. The energy spectra are obtained by the boundary condition \(J_{\nu }(2\sqrt{2a}/b)=0\) at the origin. We illustrate the fundamental properties of the wave functions in Sect. 3. The variations of the wave functions with respect to the potential parameters *a* and *b* are studied graphically. Finally, some concluding remarks are given in Sect. 4.

## Semi-exact Solutions

The Schrödinger equation with the potential *V*(*x*) can be written as

where \(k=\sqrt{-E}\) (\(E<0\)) for the bound state solutions and the potential *V*(*x*) is given in Eq. (1).

By taking

and substituting it into (2), we have

where \(\nu ^2=8k^2/b^2>0\) and we assume \(\nu >0\) but this assumption does not affect the final conclusions. Its solutions are given by

where we take \(C=0\) since the wave function must tend to zero when \(x\rightarrow \infty \). Thus, the normalization constant is calculated as

where \(_2\tilde{F}_3\) represents regularized Hypergeometric function. In the calculation we have used the following formula [31] (it is incorrect therein)

with the condition \(\mathrm{Re}(\lambda +\mu +\nu )>-1\).

Now, our aim is to see whether we could connect the Bessel differential equation to other differential equations and then find the possibility to find the analytical expression of the energy spectrum. To this end, let us take

and then we have

Its solutions are given by

where \(\,_{1}F_{1}(a,b,x)\) denotes the confluent hypergeometric function. From the quantization rule \(a=\nu +1/2=-n\), we have

Unfortunately, the energy spectra obtained in this way are *incorrect* since they are only related with one parameter *b*. On the contrary, the energy spectra should be closely related to all potential parameters *a* and *b*. The energy spectra obtained by this way do not correspond to any discrete bound states. Now, let us study the energy spectra from the behaviours of the wave functions (4). Considering the behaviours of the wave functions at origin and at the infinity, one has \(\psi (z)=0\) when \(x\rightarrow \infty \), but the eigenvalues were obtained from the boundary condition \(x\rightarrow 0\) [3, 25]

from which we are able to calculate the energy spectra for some given potential parameters *a* and *b*. It is found that the results obtained by this way are in good agreement with the numerical results as done in our previous study [26,27,28,29,30]. The corresponding calculations of the energy spectra are listed in Tables I and II with respect to the variations of the potential parameters *a* and *b*. It is shown from Table 1 that the absolute values of energy spectra become large as the parameter *a* increases, while they become small as the parameter *b* increases. Before ending this section, we give a useful remark on the energy spectra. For example, for given parameters \(a=10\) and \(b=0.1\) we notice that the energy spectrum is calculated as \(\epsilon _{1}=-8.23555\) by using the boundary condition (11) but it is evaluated as \(-8.23467\) through numerical calculation. Therefore, we can say that the energy spectra calculated by Eq. (11) are reliable. The very slight difference is from the fact that the wave function at the origin should be very limit and small but not exactly be zero.

## Property of the Wave Functions

Let us study the fundamental property of the wave functions. Since the depth of the potential well increases with the parameter *a* and the potential decays more quickly with the increasing parameter *b* as shown in Fig. 1, we find that the wave peak of the wave function will move towards the origin when the potential parameter *a* increases as illustrated in Fig. 2. The similar phenomenon is also for the increasing *b* (see Fig. 3). However, for example \(b>0. 8\) for a given parameter *a*, say \(a=35\) here in Table 2, some bound state solutions are not permissable. This means that the energy spectra are very sensitive to the potential parameter *b* since this parameter makes the potential decay very quickly. Its increasing makes the width of the potential well narrower as displayed in Panel (b) of Fig. 1.

More intuitively, the potential *V*(*x*) provides a force \(a\,b\,e^{-b\,x}\in [0, ab]\) (from \(-dV(x)/dx\)), which is always positive since the potential parameters *a* and *b* are taken positive in this work. The force becomes more attractive as the parameter *a* tends to infinity, causing the peaks of the associated wave functions to move toward the origin. However, this force becomes very limited as the potential parameter *b* tends to infinity since the exponential term \(e^{-b\,x}\) goes to zero in this case.

## Concluding Remarks

In this work we have presented the semi-exact solutions of a class of exponential potential model (1), which are given by the Bessel functions. However, the energy spectra \(E=-b^2(n+1/2)^2/8\) which were obtained from the quantization condition do not correspond to any discrete bound states. The energy levels have to be calculated by the boundary condition \(J_{\nu }(2\sqrt{2a}/b)=0\) at the origin. It is found that the energy spectra obtained by this condition are in good agreement with the numerical results. The basic properties of the wave functions are illustrated through varying the potential parameters *a*, *b*. It is found that they are pull back to the origin when the potential parameter *a* increases.

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## Acknowledgements

We would like to thank the kind referee for making invaluable and positive criticisms and suggestions which have improved the manuscript greatly. This work is supported by project 20200981-SIP-IPN, COFAA-IPN, Mexico and partially by the CONACYT project under grant No. 288856.

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Shi, YJ., Sun, GH., Ortigoza, R.S. *et al.* The Exact Solutions of a Class of Monotonic Exponential Potential Model.
*Few-Body Syst* **62, **11 (2021). https://doi.org/10.1007/s00601-021-01595-3

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