# Cornell Potential: A Neural Network Approach

Authors: Halil Mutuk
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

We solved Schrödinger equation with Cornell potential (Coulomb-plus-linear potential) by using neural network approach. Four different cases of Cornell potential for different potential parameters were used without a physical relevance. Besides that charmonium, bottomonium and bottom-charmed spin-averaged spectra were also calculated. Obtained results are in good agreement with the reference studies and available experimental data.


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## Body

## 1. Introduction

The Schrödinger equation with Coulomb-plus-linear potential (Cornell potential) has received a great deal of attention as an important nonrelativistic model for the study of quark-antiquark systems, namely, mesons [1–18]. Especially Coulomb-plus-linear potential was the first potential model to study heavy quarkonium systems and inspired other phenomenological models. The quarkonium spectrum is a substantial field to improve our understanding about nature of QCD. The Cornell potential reads as(1)Vr=-ar+br,where a and b are some constants. At small quark-antiquark distances, the Cornell potential goes like ≈1/r which is known as Coulomb-like term and at large distances it goes as ≈r, which is known as linear term. The Coulomb-like term arises from one-gluon exchange and linear term presumably arises from higher order effects [19]. Up to now, the linear term has not been calculated from the first principles of QCD.Aside from its physical relevance, the Schrödinger equation with Coulomb-plus-linear potential has been studied with pure mathematical techniques. Hall used the method of potential envelopes to construct general upper and lower bounds on the eigenvalues of the Hamiltonian representing a single particle in Coulomb-plus-linear potential [20]. In [21], Chaudhuri et al. used Hill determinant method to bound state eigenvalue problem with Coulomb-plus-linear potential. Plante and Antippa solved the Schrödinger equation for a quark-antiquark system interacting via a Coulomb-plus-linear potential and obtained the wave functions as power series, with their coefficients given in terms of the combinatorics functions [22]. In [23], the authors presented a numerically precise treatment of the Crank–Nicolson method with an imaginary time evolution operator in order to solve the Schrödinger equation. Although such models have been studied to solve Schrödinger equations numerically or analytically, exact solutions of Schrödinger equation with Cornell potential are still unknown.Artificial neural networks (ANNs) can be used as an alternative method to solve both ordinary and partial differential equations. ANN is a parallel distributed processor in which numerous numbers of simply designed computing units exist. These units are calledneurons. Its massive connections between neurons can be used for storing various types of informations, particularly the ones classified asknowledge orexperience [24, 25]. This information is acquired as interneuron connection strengths, or synaptic weights.Especially with the work of [26, 27], ANNs are being used widely for solving ordinary and partial differential equations. ANNs have many advantages compared to existing semianalytic and numerical methods. The main advantages of using neural networks to solve differential equations can be stated as follows [28]:(i)
Neural networks (NNs) are universal functions approximators which are useful in solving differential equations.(ii)
Trial solutions of artificial neural networks involve a single independent variable regardless of the dimension of the problem.(iii)
The approximate solutions are continuous over all the domain of integration. In contrast, the numerical methods provide solutions only over discrete points and the solution between these points must be interpolated.(iv)
The computational complexity does not increase considerably with the number of sampling points and with the number of dimensions in problem.Being an eigenvalue problem, Schrödinger equation had got attention by ANNs at the beginning of 90s. In [29], the authors taught a neural network the solution of the two dimensional Schrödinger equation for some model potential energy functions. In work of Androsiuk et al. [30], they presented computer simulations of a neural network capable of learning to perform transformations generated by the Schrödinger equation required to find eigenenergies of two dimensional harmonic oscillator. Sugawara presented a new approach for solving the Schrödinger equation based on genetic algorithm and artificial neural network. The method was tested in the calculation of one-dimensional harmonic oscillator and other model potentials such as Morse potential [31].In the present paper, we solve Schrödinger equation with Coulomb-plus-linear potential via ANN system. In Section2, we introduce the formalism of ANN method and describe how it can be applied to the quantum mechanical calculations. In Section 3, we give and discuss numerical results for the eigenvalues of Coulomb-plus-linear potential and in Section 4 we summarize our findings.


## 2. Formalism

The neural network (NN) is constructed as a model of simply designed computing unit, calledneuron. Figure 1 illustrates a simple model of a single neuron with multiple inputs and one output.Figure 1
A model of single neuron.All the input signals are summed up asz and the output signal is determined by the nonlinear activation function σ(z). In this work we use a sigmoid function(2)σz=11+e-zas an activation function since it is possible to derive all the derivatives of σ(z) in terms of itself. This differentiability is an important aspect for the Schrödinger equation. Here we use a multilayer perceptron neural network (MLPN). The architecture of MLPN is shown in Figure 2.Figure 2
A model of multilayer neural networks.This is an example of feed forward neural network. Feed forward neural networks are the most popular architectures due to their structural flexibility, good representational capabilities, and availability of a large number of training algorithm [24]. This network consists of neurons arranged in layers in which every neuron is connected to all neurons of the next layer and the information process can be in only one way, from input to the output. The input-output properties of the neurons (perceptrons in computerized systems) can be written as(3)oi=σni,(4)oj=σnj,(5)ok=σnk, where i is for input, j is for hidden, and k is for output layers. Input to the perceptrons is given as(6)ni=InputsignaltotheNN,(7)nj=∑i=1Niωijoi+θj,(8)nk=∑i=1Njωjkoj+θk. Here, Ni and Nj represent the numbers of the units belonging to input and hidden layers, ωij is the synaptic weight parameter which connects the neurons i and j, and θj represents threshold parameter for the neuron j [30]. The overall response of the network is given as(9)ok=∑j=1bnωjkσ∑i=1anωijni+θj+θk.The derivatives of ok with respect to network parameters (weights and thresholds) can be done by differentiating (9) as(10)∂ok∂ωij=ωjkσ1njni,(11)∂ok∂ωjk=σnjδkk′,(12)∂ok∂θj=ωjkσ1nj,(13)∂ok∂θk=δkk′. Neural network systems are self-training systems. Therefore parameters of training need updating in the transfer process and these equations play important roles on the learning processes of the neural networks.


### 2.1. Implementation of ANN to Quantum System

We consider the application of the ANN to a quantum mechanical calculation. We will follow the formalism which was developed in [26]. Consider the following differential equation:(14)HΨr=frwhere H is a linear operator, f(r) is a known function, and Ψ(r)=0 at the boundaries. In order to solve this differential equation a trial function can be written as(15)Ψtr=Ar+Br,λNr,p, which employs a feed forward neural network with parameter vectors p and λ to be adjusted. The parameter vector p corresponds to the weights and biases of the neural architecture. The functions A(r) and B(r,λ) should be specified in a convenient way so that Ψt(r) satisfies the boundary conditions regardless of the p and λ values. To obtain a solution for (14), the collocation method can be used and the differential equation can be transformed into a minimization problem(16)minp,λ∑iHΨtri-fri2.For Schrödinger equation, (14) takes the form(17)HΨr=ϵΨr with the boundary condition Ψ(r)=0. In this case, the trial solution can be written as(18)Ψtr=Br,λNr,p where B(r,λ)=0 at boundary conditions for a range of λ values. By discretizing the domain of the problem, it is transformed into a minimization problem with respect to the parameters p and λ(19)Ep,λ=∑iHΨtri,p,λ-ϵΨtri,p,λ2∫Ψt2dr, where E is the error function and ϵ can be computed as(20)ϵ=∫Ψt∗HΨtdr∫Ψt2dr.Let us consider a multilayer perceptron withn input units, one hidden layer with m units and one output. This multilayer perceptron looks like Figure 2 but with just one output. For a given input vector(21)r=r1,…,rn, the output of the network is(22)N=∑i=1mνiσzi, where(23)zi=∑j=1nωijrj+ui. Here, ωij denotes the weight from the input unit j to the hidden unit i, νi is the weight from the hidden unit i to the output, ui is the bias of hidden unit i, and σ(z) is the sigmoid function, given in (2). The derivatives of network output can be defined as(24)∂kN∂rjk=∑i=1mνiωijkσik where σi=σ(zi) and σ(k) is the kth order derivative of the sigmoid.Once the derivatives of the error with respect to the network parameters have been defined, any minimization technique can be carried out. In this work, we used a feed forward neural network with a back propagation algorithm.By employing this approach it is possible to obtain energy eigenvalues of the Cornell potential. Before obtaining Cornell potential eigenvalues, it will be useful to give an example.


#### 2.1.1. A Warm up Example: Yukawa Potential

The Yukawa potential has an important role in various branches of physics. For example, in plasma physics, it is known as the Debye-Hückel potential; in solid-state physics and atomic physics, it is called the Thomas-Fermi or the screened Coulomb potential and has a role in the nucleon-nucleon interaction arising out of the one-pion-exchange mechanism in nuclear physics [34]. The Yukawa potential reads as(25)Vr=-Are-αr where the parameters A and α are given by different expressions, depending on the problem under consideration. Yukawa potential does not admit an exact solution and therefore various approximate analytic and numerical methods. The ground state energy for ħ=m=1, A=2, and g=0.002 in α=gA reported by [34] is E=-0.996006, by [35] is E=-0.99601, and by [36] is E=-0.9960.We parametrize trial function as(26)ϕtx=e-βx2Nx,u,w,v,β>0 where N denotes a feed forward artificial neural network with one hidden layer and m sigmoid hidden units(27)Nx,u,w,v=∑j=1mνjσωjx+uj. The minimization problem is(28)∑iHϕtxi-ϵϕtxi2∫ϕtx2dx. The energy eigenvalue obtained with this scheme is E=-0.996000170000301 which is in good agreement with the numerical results in first five digits.


## 2.1. Implementation of ANN to Quantum System

We consider the application of the ANN to a quantum mechanical calculation. We will follow the formalism which was developed in [26]. Consider the following differential equation:(14)HΨr=frwhere H is a linear operator, f(r) is a known function, and Ψ(r)=0 at the boundaries. In order to solve this differential equation a trial function can be written as(15)Ψtr=Ar+Br,λNr,p, which employs a feed forward neural network with parameter vectors p and λ to be adjusted. The parameter vector p corresponds to the weights and biases of the neural architecture. The functions A(r) and B(r,λ) should be specified in a convenient way so that Ψt(r) satisfies the boundary conditions regardless of the p and λ values. To obtain a solution for (14), the collocation method can be used and the differential equation can be transformed into a minimization problem(16)minp,λ∑iHΨtri-fri2.For Schrödinger equation, (14) takes the form(17)HΨr=ϵΨr with the boundary condition Ψ(r)=0. In this case, the trial solution can be written as(18)Ψtr=Br,λNr,p where B(r,λ)=0 at boundary conditions for a range of λ values. By discretizing the domain of the problem, it is transformed into a minimization problem with respect to the parameters p and λ(19)Ep,λ=∑iHΨtri,p,λ-ϵΨtri,p,λ2∫Ψt2dr, where E is the error function and ϵ can be computed as(20)ϵ=∫Ψt∗HΨtdr∫Ψt2dr.Let us consider a multilayer perceptron withn input units, one hidden layer with m units and one output. This multilayer perceptron looks like Figure 2 but with just one output. For a given input vector(21)r=r1,…,rn, the output of the network is(22)N=∑i=1mνiσzi, where(23)zi=∑j=1nωijrj+ui. Here, ωij denotes the weight from the input unit j to the hidden unit i, νi is the weight from the hidden unit i to the output, ui is the bias of hidden unit i, and σ(z) is the sigmoid function, given in (2). The derivatives of network output can be defined as(24)∂kN∂rjk=∑i=1mνiωijkσik where σi=σ(zi) and σ(k) is the kth order derivative of the sigmoid.Once the derivatives of the error with respect to the network parameters have been defined, any minimization technique can be carried out. In this work, we used a feed forward neural network with a back propagation algorithm.By employing this approach it is possible to obtain energy eigenvalues of the Cornell potential. Before obtaining Cornell potential eigenvalues, it will be useful to give an example.


### 2.1.1. A Warm up Example: Yukawa Potential

The Yukawa potential has an important role in various branches of physics. For example, in plasma physics, it is known as the Debye-Hückel potential; in solid-state physics and atomic physics, it is called the Thomas-Fermi or the screened Coulomb potential and has a role in the nucleon-nucleon interaction arising out of the one-pion-exchange mechanism in nuclear physics [34]. The Yukawa potential reads as(25)Vr=-Are-αr where the parameters A and α are given by different expressions, depending on the problem under consideration. Yukawa potential does not admit an exact solution and therefore various approximate analytic and numerical methods. The ground state energy for ħ=m=1, A=2, and g=0.002 in α=gA reported by [34] is E=-0.996006, by [35] is E=-0.99601, and by [36] is E=-0.9960.We parametrize trial function as(26)ϕtx=e-βx2Nx,u,w,v,β>0 where N denotes a feed forward artificial neural network with one hidden layer and m sigmoid hidden units(27)Nx,u,w,v=∑j=1mνjσωjx+uj. The minimization problem is(28)∑iHϕtxi-ϵϕtxi2∫ϕtx2dx. The energy eigenvalue obtained with this scheme is E=-0.996000170000301 which is in good agreement with the numerical results in first five digits.


## 2.1.1. A Warm up Example: Yukawa Potential

The Yukawa potential has an important role in various branches of physics. For example, in plasma physics, it is known as the Debye-Hückel potential; in solid-state physics and atomic physics, it is called the Thomas-Fermi or the screened Coulomb potential and has a role in the nucleon-nucleon interaction arising out of the one-pion-exchange mechanism in nuclear physics [34]. The Yukawa potential reads as(25)Vr=-Are-αr where the parameters A and α are given by different expressions, depending on the problem under consideration. Yukawa potential does not admit an exact solution and therefore various approximate analytic and numerical methods. The ground state energy for ħ=m=1, A=2, and g=0.002 in α=gA reported by [34] is E=-0.996006, by [35] is E=-0.99601, and by [36] is E=-0.9960.We parametrize trial function as(26)ϕtx=e-βx2Nx,u,w,v,β>0 where N denotes a feed forward artificial neural network with one hidden layer and m sigmoid hidden units(27)Nx,u,w,v=∑j=1mνjσωjx+uj. The minimization problem is(28)∑iHϕtxi-ϵϕtxi2∫ϕtx2dx. The energy eigenvalue obtained with this scheme is E=-0.996000170000301 which is in good agreement with the numerical results in first five digits.


## 3. Numerical Results and Discussion

The radial part of the Schrödinger equation for Coulomb-plus-linear potential is(29)d2ϕrdr2+2mħ2E-ll+12mr2ħ2+Vrϕr=0where V(r)=-a/r+br and m=ħ=1. We trained the network with 200 equidistance points with m=10 in the interval -6≤r≤6 and solved the Schrödinger equation in four cases.


### 3.1. Case 1:V(r)=-1/r+r

PuttingV(r)=-1/r+r into (29) we get (30)-12d2ϕrdr2--1r+r-ll+12r2ϕr=Eϕr.The Hamiltonian of this equation is(31)H=-12d2dr2--1r+r-ll+12r2. By applying the procedure which was mentioned in the previous section, we get eigenvalues of this potential. In Tables 1 and 2, we present eigenvalues of the potential V(r)=-1/r+r for n and l states, respectively.Table 1
Comparison of eigenvalues ofV(r)=-1/r+r for n states.
l n E n 0 [13] [32] 0 0 1.3978757858276367 1.397875641660 1.39788 1 3.4750833511352539 3.475086545396 3.47509 2 5.0329142808914185 5.032914359536 5.03291 3 6.3701504468917847 6.370149125486 6.37015 4 7.5749331712722778 7.57493264059 7.57493 5 8.6879128217697144 8.687914590401 8.68791Table 2
Comparison of eigenvalues ofV(r)=-1/r+r for l states.
n l E 0 l [13] [32] 0 0 1.3978757858276367 1.397875641660 1.39788 1 2.8256469964981079 2.825646640704 2.82565 2 3.8505786657333374 3.850580006803 3.85058 3 4.7267490625381470 4.726752007096 4.72675 4 5.5169814825057983 5.516979644329 5.51698 5 6.2483960390090942 6.248395598411 6.24840It can be seen from Tables1 and 2 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies.


### 3.2. Case 2:V(r)=-1/r+0.01r

The Hamiltonian for this potential is(32)H=-12d2dr2--1r+0.01r-ll+12r2. In Tables 3 and 4, we present eigenvalues of the potential V(r)=-1/r+0.01r for n and l states, respectively.Table 3
Comparison of eigenvalues ofV(r)=-1/r+0.01r for n states.
l n E n 0 [13] [32] 0 0 - 0.221030285825463 - 0.221030563404 - 0.221031 1 0.0347223105947489 0.034722241998 0.0347222 2 0.1419130671269874 0.141913022811 0.141913 3 0.2202872073658741 0.220287171811 0.220287 4 0.2861103278041072 0.344602792592a 0.286111 5 0.3446028206061204 0.448055673514a 0.344602 a In a private communication with R. L. Hall, corresponding author of [13], he mentioned there was a copy-typing error in the order with these values. E40 is absent in their work.Table 4
Comparison of eigenvalues ofV(r)=-1/r+0.01r for l states.
n l E 0 l [13] [32] 0 0 - 0.221030285825463 - 0.221030563404 - 0.221031 1 0.0174005540419951 0.017400552510 0.0174006 2 0.1024748170092927 0.102472150415 0.102472 3 0.1598337896864147 0.159830894613 0.159831 4 0.2062381776256258 0.206238109687 0.206238 5 0.2466820720535486 0.246682072100 0.246681It can be seen from Tables3 and 4 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies.


### 3.3. Case 3:V(r)=-1/r+100r

The Hamiltonian for this potential is(33)H=-12d2dr2--1r+100r-ll+12r2. In Tables 5 and 6, we present eigenvalues of the potential V(r)=-1/r+100r for n and l states, respectively.Table 5
Comparison of eigenvalues ofV(r)=-1/r+100r for n states.
l n E n 0 [13] [32] 0 0 46.402327405478400 46.402258652779 46.40221 1 85.339352424879541 85.339271687574 85.3393 2 116.72879854879542 116.728692980119 116.729 3 144.31559756874218 144.315456241781 144.315 4 169.46073150674210 169.460543870657 169.461 5 192.85053644685147 192.850291861086 192.851Table 6
Comparison of eigenvaluesV(r)=-1/r+100r for l states.
n l E 0 l [13] [32] 0 0 46.402327405478400 46.402258652779 46.4022 1 70.018265184426244 70.016058921076 70.0161 2 89.717285390446149 89.715370910984 89.7154 3 107.36655900853525 107.334329106273 107.334 4 123.56178707194739 123.561985764157 123.562 5 138.7608054659801 138.761138633388 138.761It can be seen from Tables5 and 6 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies. In this case, numerical results of orbitally excited sates can differ in first two or three digits. The reason for this difference can be the dominance of the linear part of the potential compared to the Coulomb part.


### 3.4. Case 4:V(r)=-a/r+r

The Hamiltonian for this potential is(34)H=-12d2dr2--ar+r-ll+12r2. In Table 7, we present ground state eigenvalues of the potential V(r)=-a/r+r for varying a values. It can be seen from Table 7 that obtained results are in good agreement with the given studies.Table 7
Comparison of ground state eigenvalues forV(r)=-a/r+r.
a E 00 [1] [13] a E 00 [9] [13] 0.2 2.16731633527814569 2.167316 2.167316208772717 0.1 2.25367805603652147 2.253678 2.253678098810761 0.4 1.98850421340265894 1.988504 1.988503899750869 0.3 2.07894965124100145 2.078949 2.078949440194840 0.6 1.80107438954743241 1.801074 1.801073805646947 0.5 1.89590541258401687 1.895904 1.895904238476994 0.8 1.60440950560142422 1.604410 1.604408543236585 0.7 1.70393557501874021 1.703935 1.703934818031980 1.0 1.39787578582763672 1.397877 1.397875641659907 0.9 1.50241669742698608 1.502415 1.502415495453739 1.2 1.18083812306874152 1.180836 1.180833939744787 1.1 1.29071042316348541 1.290709 1.290708615983606 1.4 0.95264360884520136 0.952644 0.952640495218560 1.3 1.06817386102041328 1.068171 1.068171244486971 1.6 0.71266200850041624 0.712662 0.712657680461034 1.5 0.83416589280625410 0.834162 0.834162211049953 1.8 0.46026600634169799 0.460266 0.460260113873608 1.7 0.58805423050569871 0.588049 0.588049168557953


### 3.5. Heavy Quarkonium Spectra

Quarkonium systems are an ideal area for clarifying our understanding of QCD. They probe nearly all the energy regimes of QCD from high energy region to low energy region. In the high energy region, an expansion of coupling constant is possible and perturbative QCD is applicable. In the low energy region, such an expansion is not possible in the coupling constant and therefore nonperturbative methods need to be used. Besides that nonrelativistic QCD (NRQCD) approximation is also used for spectroscopy, decay, and production of heavy quarkonium [37, 38]. For an overview of NRQCD, see [39].In this section we obtained spin-averaged mass spectra charmonium, bottomonium, and bottom-charmed system by solving nonrelativistic Schrödinger equation. It is possible to obtain full spectra for quarkonium systems including relativistic effects, spin-spin, and spin-orbit interactions. Since most of these contributions are really small compared to the given potential, even by neglecting those effects one can find results that are close to the experimental data.The related Cornell potential is [40](35)Vr=Ar-κr with(36)A=0.191GeV2κ=0.472mb=4.7485GeVmc=1.3205GeV.In Tables8, 9, and 10, charmonium, bottomonium, and bottom-charmed spectra are presented, respectively.Table 8
Charmonium spectra. All results are in GeV.
State This work [32] Exp. [33] 1S 3.098 3.096 3.097 2S 3.688 3.672 3.686 3S 4.029 4.085 4.039 1P 3.516 3.521 3.511 2P 3.925 3.951 3.927 3P 4.301 4.310 - 1D 3.779 3.800 3.770Table 9
Bottomonium spectra. All results are in GeV.
State This work [32] Exp. [33] 1S 9.460 9.462 9.460 2S 10.026 10.027 10.023 3S 10.354 10.361 10.355 4S 10.572 10.624 10.579 1P 9.891 9.963 9.899 2P 10.258 10.299 10.260 3P 10.518 10.564 10.512 1D 10.156 10.209 10.164Table 10
Bottom-Charmed spectra. All results are in GeV.
State This work [32] Exp. [33] 1S 6.274 6.362 6.275 2S 6.839 6.911 6.842 3S 7.245 7.284 4S 7.522 7.593 1P 6.743 6.792 2P 7.187 7.178 3P 7.467 7.494 1D 7.046 7.051As can be seen in Tables8, 9, and 10 charmonium, bottomonium, and bottom-charmed spectra of ANN agree well with given reference and available experimental data.


## 3.1. Case 1:V(r)=-1/r+r

PuttingV(r)=-1/r+r into (29) we get (30)-12d2ϕrdr2--1r+r-ll+12r2ϕr=Eϕr.The Hamiltonian of this equation is(31)H=-12d2dr2--1r+r-ll+12r2. By applying the procedure which was mentioned in the previous section, we get eigenvalues of this potential. In Tables 1 and 2, we present eigenvalues of the potential V(r)=-1/r+r for n and l states, respectively.Table 1
Comparison of eigenvalues ofV(r)=-1/r+r for n states.
l n E n 0 [13] [32] 0 0 1.3978757858276367 1.397875641660 1.39788 1 3.4750833511352539 3.475086545396 3.47509 2 5.0329142808914185 5.032914359536 5.03291 3 6.3701504468917847 6.370149125486 6.37015 4 7.5749331712722778 7.57493264059 7.57493 5 8.6879128217697144 8.687914590401 8.68791Table 2
Comparison of eigenvalues ofV(r)=-1/r+r for l states.
n l E 0 l [13] [32] 0 0 1.3978757858276367 1.397875641660 1.39788 1 2.8256469964981079 2.825646640704 2.82565 2 3.8505786657333374 3.850580006803 3.85058 3 4.7267490625381470 4.726752007096 4.72675 4 5.5169814825057983 5.516979644329 5.51698 5 6.2483960390090942 6.248395598411 6.24840It can be seen from Tables1 and 2 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies.


## 3.2. Case 2:V(r)=-1/r+0.01r

The Hamiltonian for this potential is(32)H=-12d2dr2--1r+0.01r-ll+12r2. In Tables 3 and 4, we present eigenvalues of the potential V(r)=-1/r+0.01r for n and l states, respectively.Table 3
Comparison of eigenvalues ofV(r)=-1/r+0.01r for n states.
l n E n 0 [13] [32] 0 0 - 0.221030285825463 - 0.221030563404 - 0.221031 1 0.0347223105947489 0.034722241998 0.0347222 2 0.1419130671269874 0.141913022811 0.141913 3 0.2202872073658741 0.220287171811 0.220287 4 0.2861103278041072 0.344602792592a 0.286111 5 0.3446028206061204 0.448055673514a 0.344602 a In a private communication with R. L. Hall, corresponding author of [13], he mentioned there was a copy-typing error in the order with these values. E40 is absent in their work.Table 4
Comparison of eigenvalues ofV(r)=-1/r+0.01r for l states.
n l E 0 l [13] [32] 0 0 - 0.221030285825463 - 0.221030563404 - 0.221031 1 0.0174005540419951 0.017400552510 0.0174006 2 0.1024748170092927 0.102472150415 0.102472 3 0.1598337896864147 0.159830894613 0.159831 4 0.2062381776256258 0.206238109687 0.206238 5 0.2466820720535486 0.246682072100 0.246681It can be seen from Tables3 and 4 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies.


## 3.3. Case 3:V(r)=-1/r+100r

The Hamiltonian for this potential is(33)H=-12d2dr2--1r+100r-ll+12r2. In Tables 5 and 6, we present eigenvalues of the potential V(r)=-1/r+100r for n and l states, respectively.Table 5
Comparison of eigenvalues ofV(r)=-1/r+100r for n states.
l n E n 0 [13] [32] 0 0 46.402327405478400 46.402258652779 46.40221 1 85.339352424879541 85.339271687574 85.3393 2 116.72879854879542 116.728692980119 116.729 3 144.31559756874218 144.315456241781 144.315 4 169.46073150674210 169.460543870657 169.461 5 192.85053644685147 192.850291861086 192.851Table 6
Comparison of eigenvaluesV(r)=-1/r+100r for l states.
n l E 0 l [13] [32] 0 0 46.402327405478400 46.402258652779 46.4022 1 70.018265184426244 70.016058921076 70.0161 2 89.717285390446149 89.715370910984 89.7154 3 107.36655900853525 107.334329106273 107.334 4 123.56178707194739 123.561985764157 123.562 5 138.7608054659801 138.761138633388 138.761It can be seen from Tables5 and 6 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies. In this case, numerical results of orbitally excited sates can differ in first two or three digits. The reason for this difference can be the dominance of the linear part of the potential compared to the Coulomb part.


## 3.4. Case 4:V(r)=-a/r+r

The Hamiltonian for this potential is(34)H=-12d2dr2--ar+r-ll+12r2. In Table 7, we present ground state eigenvalues of the potential V(r)=-a/r+r for varying a values. It can be seen from Table 7 that obtained results are in good agreement with the given studies.Table 7
Comparison of ground state eigenvalues forV(r)=-a/r+r.
a E 00 [1] [13] a E 00 [9] [13] 0.2 2.16731633527814569 2.167316 2.167316208772717 0.1 2.25367805603652147 2.253678 2.253678098810761 0.4 1.98850421340265894 1.988504 1.988503899750869 0.3 2.07894965124100145 2.078949 2.078949440194840 0.6 1.80107438954743241 1.801074 1.801073805646947 0.5 1.89590541258401687 1.895904 1.895904238476994 0.8 1.60440950560142422 1.604410 1.604408543236585 0.7 1.70393557501874021 1.703935 1.703934818031980 1.0 1.39787578582763672 1.397877 1.397875641659907 0.9 1.50241669742698608 1.502415 1.502415495453739 1.2 1.18083812306874152 1.180836 1.180833939744787 1.1 1.29071042316348541 1.290709 1.290708615983606 1.4 0.95264360884520136 0.952644 0.952640495218560 1.3 1.06817386102041328 1.068171 1.068171244486971 1.6 0.71266200850041624 0.712662 0.712657680461034 1.5 0.83416589280625410 0.834162 0.834162211049953 1.8 0.46026600634169799 0.460266 0.460260113873608 1.7 0.58805423050569871 0.588049 0.588049168557953


## 3.5. Heavy Quarkonium Spectra

Quarkonium systems are an ideal area for clarifying our understanding of QCD. They probe nearly all the energy regimes of QCD from high energy region to low energy region. In the high energy region, an expansion of coupling constant is possible and perturbative QCD is applicable. In the low energy region, such an expansion is not possible in the coupling constant and therefore nonperturbative methods need to be used. Besides that nonrelativistic QCD (NRQCD) approximation is also used for spectroscopy, decay, and production of heavy quarkonium [37, 38]. For an overview of NRQCD, see [39].In this section we obtained spin-averaged mass spectra charmonium, bottomonium, and bottom-charmed system by solving nonrelativistic Schrödinger equation. It is possible to obtain full spectra for quarkonium systems including relativistic effects, spin-spin, and spin-orbit interactions. Since most of these contributions are really small compared to the given potential, even by neglecting those effects one can find results that are close to the experimental data.The related Cornell potential is [40](35)Vr=Ar-κr with(36)A=0.191GeV2κ=0.472mb=4.7485GeVmc=1.3205GeV.In Tables8, 9, and 10, charmonium, bottomonium, and bottom-charmed spectra are presented, respectively.Table 8
Charmonium spectra. All results are in GeV.
State This work [32] Exp. [33] 1S 3.098 3.096 3.097 2S 3.688 3.672 3.686 3S 4.029 4.085 4.039 1P 3.516 3.521 3.511 2P 3.925 3.951 3.927 3P 4.301 4.310 - 1D 3.779 3.800 3.770Table 9
Bottomonium spectra. All results are in GeV.
State This work [32] Exp. [33] 1S 9.460 9.462 9.460 2S 10.026 10.027 10.023 3S 10.354 10.361 10.355 4S 10.572 10.624 10.579 1P 9.891 9.963 9.899 2P 10.258 10.299 10.260 3P 10.518 10.564 10.512 1D 10.156 10.209 10.164Table 10
Bottom-Charmed spectra. All results are in GeV.
State This work [32] Exp. [33] 1S 6.274 6.362 6.275 2S 6.839 6.911 6.842 3S 7.245 7.284 4S 7.522 7.593 1P 6.743 6.792 2P 7.187 7.178 3P 7.467 7.494 1D 7.046 7.051As can be seen in Tables8, 9, and 10 charmonium, bottomonium, and bottom-charmed spectra of ANN agree well with given reference and available experimental data.


## 4. Summary

In this paper, we applied ANN method to deal with the solution of the Schrödinger equation with Coulomb-plus-linear potential. This potential belongs to the nonsolvable potentials class which has an exactly/analytically solvable part, together with a modifying term. We obtained the eigenvalues of Schrödinger equation and spin-averaged mass spectra of charmonium, bottomonium, and bottom-charmed systems. The obtained eigenvalues and heavy quarkonium spectra are in agreement with the theoretical studies and available experimental data.The feed forward ANNs method has a good property of function approximation. A function approximation problem is to select or find a function among a well-defined functions set that closely matches a target function in a task specific way. This form employs a feed forward neural network as the basic approximation element, whose parameters (weights and biases) are adjusted to minimize an appropriate error function. In this study, the wave function is represented by the feed forward ANN and its inputs are taken as coordinate values. A trial solution is written as a feed forward neural network which contains adjustable parameters (weights and biases) and eigenvalue is refined to the known solutions by training the neural network.This study would be useful for the exact or quasiexact spectra of a few body systems. It is also possible in principle to handle many body problems but such problems will impose much more heavier computational load and other difficulties such as convergence of sigmoid functions [41] and availability of hardware.



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