# Dirac Equation under Scalar, Vector, and Tensor Cornell Interactions

Authors: H. Hassanabadi; E. Maghsoodi; S. Zarrinkamar; H. Rahimov
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

Spin and pseudospin symmetries of Dirac equation are solved under scalar, vector, and tensor interactions for arbitrary quantum number via the analytical ansatz approach. The spectrum of the system is numerically reported for typical values of the potential parameters.


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## Body

## 1. Introduction

No doubt, the Cornell potential (alternatively called Funnel in the literature) is, if not the best, among the most appealing interactions in particle physics. The Cornell potential contains a confining term besides the Coulomb interaction and has successfully accounted for the particle physics data [1]. Unfortunately, to our best knowledge, the potential does not possess exact solutions under all common equations of quantum mechanics, that is, the nonrelativistic Schrödinger equation, and relativistic Dirac, Klein-Gordon, Proca, and Duffin-Kemmer-Petiau (DKP) equations. Here, we focus on the relativistic symmetries of Dirac equation, that is, spin and pseudospin symmetries, which provide a reliable theoretical basis for hadronic and nuclear spectroscopy [2–12]. This has motivated many studies under various interactions within the past two decades ([13–18] and many references therein). Nevertheless, none of these papers has investigated the symmetry limits under the Cornell potential. This is definitely due to the complicated nature of the resulting differential equation which cannot be solved by common analytical techniques of quantum mechanics such as the supersymmetry quantum mechanics (SUSYQ), Lie groups, Nikiforov-Uvarov (NU) technique, and point canonical transformations, In our study, we make use of the ansatz approach to deal with this complicated equation. A survey on the application of this technique to other wave equations including Schrödinger, spinless-Salpeter, Dirac, Klein-Gordon, and DKP equations can be found in [19–27]. We organize the study as follows. In the first step, we review the most essential equations of the symmetry limits. We next propose a physical ansatz solution to the equation and, in a systematic manner, calculate the spectrum of the system for any arbitrary state. To provide a better understanding of the solutions, we provide some numerical data for the spectrum as well.


## 2. Dirac Equation Including Tensor Coupling

In spherical coordinates, Dirac equation with both scalar potentialS(r) and vector potential V(r) is expressed as [2, 3]
(2.1)[α→·p→+β(M+S(r))-iβα→·r^U(r)]ψ(r→)=[E-V(r)]ψ(r→),
where E is the relativistic energy of the system; α and β are the 4×4 Dirac matrices and p→=-i∇→ stands for the momentum operator. For a particle in a spherical field, the total angular momentum operator J→ and spin-orbit matrix operator K=(σ→·L→+1), where σ and L are, respectively, the Pauli matrix and orbital angular momentum, commute with the Dirac Hamiltonian. The eigenvalues of K are  κ=-(j+1/2) for the aligned spin (s1/2,p3/2, etc.) and κ=(j+1/2) for the unaligned spin (p1/2,d3/2, etc.). The complete set of the conservative quantities can be chosen as (H,K,J2,Jz). As shown in [13], the Dirac spinor is considered as
(2.2)ψnk(r→)=(fnk(r→)gnk(r→))=(Fnk(r→)rYjmℓ(θ,φ)iGnk(r→)rYjmℓ~(θ,φ)),
where Fnk(r→) and Gnk(r→) are the radial wave functions of the upper and lower components, respectively, and Yjmℓ(θ,φ) and Yjmℓ~(θ,φ), respectively, stand for spin and pseudospin spherical harmonics coupled to the angular momentum j.m  is the projection of  the angular momentum on the z-axis. The orbital angular momentum quantum numbers ℓ and ℓ~ refer to the upper and lower components, respectively. The quasidegenerate doublet structure can be expressed in terms of pseudospin angular momentum s=1/2 and pseudoorbital angular momentum ℓ~, which is defined as ℓ~=ℓ+1 for aligned spin j=ℓ~-1/2 and ℓ~=ℓ-1 for unaligned spin j=ℓ~+1/2. As shown in [2, 3], substitution of (2.2) into (2.1) yields the following two-coupled differential equations:
(2.3)(ddr+κr-U(r))Fnk(r)=(M+Enk-V(r)+S(r))Gnk(r),(ddr-κr+U(r))Gnk(r)=(M-Enk+V(r)+S(r))Fnk(r),
which simply give
(2.4){d2dr2-κ(κ+1)r2+2κrU(r)-dU(r)dr-U2(r)+dΔ(r)/drM+Enκ-Δ(r)(ddr+κr-U(r))}Fnκ(r)=(M+Enκ-Δ(r))(M-Enκ+Σ(r))Fnκ(r),(2.5){d2dr2-κ(κ-1)r2+2κrU(r)+dU(r)dr-U2(r)-dΣ(r)/drM-Enκ+Σ(r)(ddr-κr+U(r))}Gnκ(r)=(M+Enκ-Δ(r))(M-Enκ+Σ(r))Gnκ(r),
where, as the notation indicates, Δ(r)=V(r)-S(r) and Σ(r)=V(r)+S(r).


### 2.1. Pseudospin Symmetry Limit

Under the condition of the pseudospin symmetry,dΣ(r)/dr=0 or equivalently Σ(r)=Cgps=Const. [2, 3]. We choose Δ(r) as the Cornell potential:
(2.6)Δ(r)=apsr+bpsr.
For the tensor term, we consider the Cornell potential:
(2.7)U(r)=Ar+Br.
Substitution of these two terms in into (2.5) gives
(2.8){d2dr2+1r2(-κ(κ-1)+2κB-B-B2)-A2r2+(apsM-apsEnκps+apsCps)r+1r(bpsM-bpsEnκps+bpsCps)-M2-MCps+(Enκps)2-EnκpsCps+2κA+A-2ABd2dr2+1r2}×Gnκps(r)=0,
where κ=-ℓ~ and κ=ℓ~+1 for κ<0 and κ>0, respectively.


### 2.2. Spin Symmetry Limit

In the spin symmetry limitdΔ(r)/dr=0 or Δ(r)=Cgs=const. [2, 3]. As the previous section, we consider(2.9a)Σ(r)=asr+bsr,(2.9b)U(r)=Ar+Br.Substitution of the latter in (2.4) gives
(2.10){d2dr2+1r2(-κ(κ+1)+2κB+B-B2)-A2r2+(-asM-asEnκs+asCs)r+1r(-bsM-bsEnκs+bsCs)-M2+MCs+(Enκps)2-EnκsCs+2κA-A-2ABd2dr2+1r2}Fnκs(r)=0,
where κ=ℓ and κ=-ℓ-1 for κ<0 and κ>0, respectively.


## 2.1. Pseudospin Symmetry Limit

Under the condition of the pseudospin symmetry,dΣ(r)/dr=0 or equivalently Σ(r)=Cgps=Const. [2, 3]. We choose Δ(r) as the Cornell potential:
(2.6)Δ(r)=apsr+bpsr.
For the tensor term, we consider the Cornell potential:
(2.7)U(r)=Ar+Br.
Substitution of these two terms in into (2.5) gives
(2.8){d2dr2+1r2(-κ(κ-1)+2κB-B-B2)-A2r2+(apsM-apsEnκps+apsCps)r+1r(bpsM-bpsEnκps+bpsCps)-M2-MCps+(Enκps)2-EnκpsCps+2κA+A-2ABd2dr2+1r2}×Gnκps(r)=0,
where κ=-ℓ~ and κ=ℓ~+1 for κ<0 and κ>0, respectively.


## 2.2. Spin Symmetry Limit

In the spin symmetry limitdΔ(r)/dr=0 or Δ(r)=Cgs=const. [2, 3]. As the previous section, we consider(2.9a)Σ(r)=asr+bsr,(2.9b)U(r)=Ar+Br.Substitution of the latter in (2.4) gives
(2.10){d2dr2+1r2(-κ(κ+1)+2κB+B-B2)-A2r2+(-asM-asEnκs+asCs)r+1r(-bsM-bsEnκs+bsCs)-M2+MCs+(Enκps)2-EnκsCs+2κA-A-2ABd2dr2+1r2}Fnκs(r)=0,
where κ=ℓ and κ=-ℓ-1 for κ<0 and κ>0, respectively.


## 3. The Ansatz Solution

### 3.1. Solution of the Pseudospin Symmetry Limit

In the previous section, we obtained a Schrödinger-like equation of the form(3.1)d2Gnκps(r)dr2+[εnκps-a1psr2-b1psr+c1psr+d1ps-κ(κ-1)r2]Gnκps(r)=0,
where
(3.2)εnκps=-M2-MCps+(Enκps)2-EnκpsCps+2κA+A-2AB,a1ps=A2,b1ps=-apsM+apsEnκps-apsCps,c1ps=bpsM-bpsEnκps+bpsCps,d1ps=2κB-B-B2.
Equation (3.1) fails to admit exact analytical solutions. Therefore, we follow the ansatz approach with the starting square:
(3.3)Gnκps(r)=fnps(r)exp(gκps(r)),
where
(3.4)fnps(r)={1,ifn=0,∏i=1n(r-αin),ifn≥1,(3.5)gκps(r)=-12αpsr2-βpsr+δpslnr,αps>0,βps>0.
By substitution of fn(r) and gκ(r) into (3.3), we find
(3.6)Gnκps′′(r)=[gκps′′(r)+gκps′2(r)+fnps′′(r)+2gκps′(r)fnps′(r)fnps(r)]Gnκps(r).
Here, we consider the case n=0. From (3.4)–(3.6) we find
(3.7)G0κps′′(r)={(αps)2r2+2αpsβpsr-αps[1+2(δps+0)]+(βps)2-2βps(δps+0)r+δps(δps-1)r2}G0κps(r).
By comparing the corresponding powers of (3.1) and (3.7), we have
(3.8)αps=a1ps,a1ps>0,βps=b1ps2a1ps,c1ps=2βps(δps+0),δps=12(1±(2κ-1)2-4d1ps)=12(1±k1ps),ε0κps=αps[1+2(δps+0)]-(βps)2,
where k1ps=(2κ-1)2-4d1ps. Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take δ from (3.8) as
(3.9)δps=12(1+k1ps).
Form (3.2), (3.8), the ground-state energy satisfies(3.10a)-M2-MCps+(E0κps)2-E0κpsCps+2κA+A-2AB-αps[1+2(δps+0)]+βps2=0
or
(3.10b)(E0κps)2-E0κpsCps-M2-MCps+2κA+A-2AB-a1ps(2+(2κ-1)2-4d1ps)+b1ps24a1ps=0,which is more compactly written as
(3.11)E0κps±=12{+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2Cps±[4κ2-4κ+1-4d1ps-4βps2Cps2-8κA-4A+8AB+4M2+4MCps+8a1ps+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2},(3.35),
where the parameter aps of potential (2.6) from (3.8) should satisfy the following restriction:
(3.12)aps=-2bpsa1ps(1+(2κ-1)2-4d1ps)=-2bpsa1ps(1+k1ps).
From (3.3), (3.4), and (3.8), the upper and lower components of the wave function are(3.13a)gκ(r)=-12a1psr2-b1ps2a1psr+12(1+(2κ-1)2-4d1ps)lnr,(3.13b)G0κps(r)=N0κr(1/2)(1+(2κ-1)2-4d1ps)exp(-12a1psr2-b1ps2a1psr),(3.13c)F0κps(r)=1M-E0κps+Cps(ddr-κr+U(r))G0κps(r).For the first node (n=1), using f1(r)=(r-α1(1)) and gκ(r) from (3.5), we arrive at
(3.14)a1psr2+b1psr-c1psr+κ(κ-1)-d1psr2-ε1κps=-αps-δpsr2+(αps)2r2+(βps)2+(δps)2r2+2αpsβpsr-2δpsαps-2βpsδpsr+-2αpsr-2βps+2δps/rr-α1(1).
Here, the consequent relations between the potential parameters and the coefficients αps, βps, δps, and α1(1) are
(3.15)αps=a1ps,βps=b1ps2a1ps,δps=12(1+k1ps),ε1κps=αps[1+2(δps+1)]-(βps)2,(c1ps-2βpsδps)α1(1)=2δps,c1ps-2βps(δps+1)=2αpsα1(1).
By solving the above equations one can find c1ps and α1(1) as(3.16a)c1ps=b1ps2a1ps((2κ-1)2-4d1ps+2)+b1ps24a1ps+2a1ps(1+(2κ-1)2-4d1ps),αpsα1(1)2+βpsα1(1)-δps=0,(3.16b)→α1(1)=12-βps+(βps)2+4αpsδpsαps=-b1ps4a1ps+b1ps216a1ps2+1+(2κ-1)2-4d1ps2a1ps.The energy eigenvalues therefore are(3.17)E1κps±=12{+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2Cps±[4a1ps4κ2-4κ+1-4d1ps-4βps2Cps2-8κA-4A+8AB+4M2+4MCps+16a1ps+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2},
where the parameter aps from (3.8) is
(3.18)aps=2a1ps(2c1ps+c1psk1ps+c1ps2+6a1ps+14a1psk1ps+10a1psk1ps2+2a1psk1ps3)-3M-4Mk1ps-Mk1ps2+3E1κps+4E1κpsk1ps+E1κpsk1ps2-3Cps-4Cpsk1ps-Cpsk1ps2.
For the upper and lower components of the wave function we thus have(3.19a)G1κps(r)=N1κ(r-α1(1))r(1/2)(1+(2κ-1)2-4d1ps)exp(-12a1psr2-b1ps2a1psr),(3.19b)F1κps(r)=1M-E1κps+Cps(ddr-κr+U(r))G1κps(r).Following the analytic iteration procedures for the second node(n=2) with f2(r)=(r-α1(2))(r-α2(2)) and gκ(r) as defined in (3.6), the relations between the potential parameters and the coefficients αps, βps, δps, α1(2), and α2(2)are
(3.20)αps=a1ps,βps=b1ps2a1ps,δps=12(1+k1ps),ε2κps=αps[1+2(δps+2)]-(βps)2,c1ps-2βps(δps+2)=2αps∑i=12αi(2),(c1ps-2βpsδps)∑i<j2αi(2)αj(2)=2δps∑i=12αi(2),[c1ps-2βps(δps+1)]∑i=12αi(2)=4αps∑i<j2αi(2)αj(2)+2(2δps+1).
The coefficients α1(2) and α2(2) are found from the constraint relations [20, 27–29]:
(3.21)αps∑i=12αi(2)2+βps∑i=12αi(2)-(2δps+1)=0,(3.22)δps∑i=12αi(2)2-(βps∑i=12αi(2)+1)∑j<k2αj(2)αk(2)-2αps∑j<k2αj(2)2αk(2)2=0.
Therefore, the energy eigenvalue in this case is
(3.23)E2κps±=12{+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2Cps±[>+4a1ps4κ2-4κ+1-4d1ps-4βps2Cps2-8κA-4A+8AB+4M2+4MCps+24a1ps+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2},
and the lower component of the wave function is
(3.24)G2κps(r)=N2κ∏i=12(r-αi(2))r(1/2)(1+(2κ-1)2-4d1ps)exp(-12a1psr2-b1ps2a1psr).


### 3.2. Solution of the Spin Symmetry Limit

In this case, our ordinary differential equation is(3.25)d2Fnκs(r)dr2+[εnκs-a1sr2-b1sr+c1sr+d1s-κ(κ+1)r2]Fnκs(r)=0,
with
(3.26)εnκs=-M2+MCs+(Enκs)2-EnκsCs+2κA-A-2AB,a1s=A2,b1s=asM+asEnκs-asCs,c1s=-bsM-bsEnκs+bsCs,d1s=2κB+B-B2,
which cannot be solved by our common exact analytical techniques. Let us propose the ansatz solution:
(3.27)Fnκs(r)=fns(r)exp(gκs(r)),
where
(3.28)fns(r)={1,ifn=0,∏i=1n(r-αin),ifn≥1,(3.29)gκs(r)=-12αsr2-βsr+δslnr,αs>0,βs>0.
By substitution of fn(r) and gκ(r) into (3.27), we find
(3.30)Fnκs′′(r)=[gκs′′(r)+gκs′2(r)+fns′′(r)+2gκs′(r)fns′(r)fns(r)]Fnκs(r).
For the case of n=0, from (3.26)–(3.29), we find
(3.31)Fnκs′′(r)={(αs)2r2+2αsβsr-αs[1+2(δs+0)]+(βs)2-2βs(δs+0)r+δs(δs-1)r2}Fnκs(r).
By comparing the corresponding powers of (3.27) and (3.33), we have
(3.32)αs=a1s,βs=b1s2a1s,a1s>0,c1s=2βs(δs+0),δs=12(1±(2κ+1)2-4d1s)=12(1±k1s),ε0κs=α[1+2(δs+0)]-(βs)2,
where k1s=(2κ+1)2-4d1s. To have physically acceptable solutions, we pick up the value
(3.33)δs=12(1+k1s).
By considering (3.26), (3.32), the first node eigenvalue satisfies
(3.34)-M2+MCs+(Enκs)2-EnκsCs+2κA-A-2AB-αs[1+2(δs+0)]+(βs)2=0,
or equivalently
(3.35)E0κs±=12{+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs±[+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs2+4M2-4MCs-8κA+4A+8AB+8a1s+4a1s4κ2+4κ+1-4d1s-4βs2+4a1s4κ2+4κ+1-4d1s-4βs2]1/2]1/2},
where the parameter as of potential (2.9a) should satisfy the restriction
(3.36)as=-2bsa1s(1+k1s).
From (3.27), (3.28), and (3.32), the upper and lower components of the wave function are
(3.37)gκ(r)=-12a1sr2-b1s2a1sr+12(1+(2κ+1)2-4d1s)lnr,(3.38)F0κs(r)=N0κr(1/2)(1+(2κ+1)2-4d1s)exp(-12a1sr2-b1s2a1sr),(3.39)G0κs(r)=1M+E0κs-Cs(ddr+κr-U(r))F0κs(r).
Secondly, for the first node (n=1), using f1(r)=(r-α1(1)) and gκ(r) from (3.28)–(3.30), our resulting equation is
(3.40)a1sr2+b1sr-c1sr+-d1s+κ(κ+1)r2-εnκs=-αs-δsr2+(αs)2r2+(βs)2+(δs)2r2+2αsβsr-2δsαs-2βsδsr+-2αsr-2βs+2δs/rr-α1(1).
The relations between the potential parameters and the coefficients αs,βs,δs, and α1(1) are
(3.41)αs=a1s,βs=b1s2a1ps,a1s>0,δs=12(1+(2κ+1)2-4d1s)=12(1+k1s),ε1κs=αs[1+2(δs+1)]-(βs)2,(c1s-2βsδs)α1(1)=2δs,c1s-2βs(δs+1)=2αsα1(1),
where c1s and α1(1) are found from (3.41) as [20, 27–29](3.42a)c1s=b1s2a1s((2κ+1)2-4d1s+2)+b1s24a1s+2a1s(1+(2κ+1)2-4d1s),(3.42b)αsα1(1)2+βsα1(1)-δs=0,
or
(3.42c)α1(1)=12-βs+(βs)2+4αsδsαs=-b1s4a1s+b1s216a1s2+1+(2κ+1)2-4d1s2a1s,which determine the corresponding energy as
(3.43)E1κs±=12{+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs±[4κ2+4κ+1-4d1s-4βs2Cs2+4M2-4MCs-8κA+4A+8AB+16a1s+4a1s4κ2+4κ+1-4d1s-4βs2]1/2},
with
(3.44)as=2a1s(2c1s+c1sk1s+c1s2+6a1s+14a1sk1s+10a1sk1s2+2a1sk1s3)3M+4Mk1s+Mk1s2+3E1κs+4E1κsk1s+E1κsk1s2-3Cs-4Csk1s-Csk1s2.
The upper and lower components of the wave function are then simply found to be(3.45a)F1κs(r)=N1κ(r-α1(1))r(1/2)(1+(2κ+1)2-4d1s)exp(-12a1sr2-b1s2a1sr),(3.45b)G1κs(r)=1M+E1κs-Cs(ddr+κr-U(r))F1κs(r).For the second node(n=2), we choose f2(r)=(r-α1(2))(r-α2(2)) and gκ(r) as defined in (3.28) and (3.29). The relations between the potential parameters and the coefficients αs,βs,δs,α1(2), and α2(2) are
(3.46)αs=a1s,βs=b1s2a1s,δs=12(1+k1s),ε2κs=αs[1+2(δs+2)]-(βs)2,c1s-2βs(δs+2)=2αs∑i=12αi(2),(c1s-2βsδs)∑i<j2αi(2)αj(2)=2δs∑i=12αi(2),[c1s-2βs(δs+1)]∑i=12αi(2)=4α∑i<j2αi(2)αj(2)+2(2δs+1),
where
(3.47)αs∑i=12αi(2)2+βs∑i=12αi(2)-(2δs+1)=0,(3.48)δs∑i=12αi(2)2-(βs∑i=12αi(2)+1)∑j<k2αj(2)αk(2)-2αs∑j<k2αj(2)2αk(2)2=0.
The energy eigenvalue therefore is
(3.49)E2κs±=12{+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs±[a1s4κ2+4κ+1-4d1s-4βs2Cs2+4M2-4MCs-8κA+4A+8AB+24a1s+4a1s4κ2+4κ+1-4d1s-4βs2]1/2}.
For the upper component of the wave function, we have
(3.50)F2κs(r)=N2κ∏i=12(r-αi(2))r(1/2)(1+(2κ+1)2-4d1s)exp(-12a1sr2-b1s2a1sr).
We have given some numerical values of the energy eigenvalues in Tables 1, 2, 3, 4, 5, and 6 for various states. For the final point, we wish to emphasize on the degeneracy-removing role of the (Cornell) tensor potential. As we already know, for vanishing tensor interaction (A=B=0), the pseudospin doublets, that is, states with quantum numbers (n,l,j=l+1/2) and (n-1,l+2,j=l+3/2) are degenerate. The degenerate states in the spin doublets are those with quantum numbers (n,ℓ,j=ℓ+1/2) and (n,ℓ,j=ℓ-1/2), where n,l,and j are the radial, the orbital, and the total angular momentum quantum numbers, respectively (see Tables 1 and 2). Our numerical data reveals that, in the pseudospin symmetry limit, the degenerate states for (A=B=0) are ns1/2,(n-1)d3/2 for ℓ~=1(ℓ=0), np3/2,(n-1)f5/2 for ℓ~=2(ℓ=1), nd5/2,(n-1)g7/2 for ℓ~=3(ℓ=2), nf7/2,(n-1)h9/2 for ℓ~=4(ℓ=3), and so forth. For spin symmetry limit and (A=B=0) one can clearly see that the degeneracy occurs in (np1/2,np3/2) for ℓ=1, (nd3/2,nd5/2) for ℓ=2,(nf5/2,nf7/2) for ℓ=3, (ng7/2,ng9/2) for ℓ=4, and so forth.Table 1
Bound state for the pseudospin symmetryb=1, M=1 fm−1, Cps=0 fm−1.
ℓ
~
n
,
κ
(
ℓ
,
j
)
E
n
κ
ps
- (fm−1)
A
=
B
=
0
A
=
0.5
,
B
=
0.2
1
0, −1
0
S
1
/
2
−0.882352941
−1.984891569
2
0, −2
0
P
3
/
2
−0.945945946
−2.471004354
3
0, −3
0
d
5
/
2
−0.969230769
−2.861584264
4
0, −4
0
f
7
/
2
−0.980198020
−3.199522676
1
1, −1
1
S
1
/
2
−0.945945946
−2.302587881
2
1, −2
1
p
3
/
2
−0.969230769
−2.710867336
3
1, −3
1
d
5
/
2
−0.980198020
−3.060956677
4
1, −4
1
f
7
/
2
−0.986206897
−3.373431922
1
1, 2
0
d
3
/
2
−0.945945946
−1.396649362
2
1, 3
0
f
5
/
2
−0.969230769
−1.411440323
3
1, 4
0
g
7
/
2
−0.980198020
−1.414060440
4
1, 5
0
h
9
/
2
−0.986206897
−1.414104092Table 2
Bound state for the spin symmetryb=1, M=1 fm−1, Cs=0 fm−1.
ℓ
n
,
κ
(
ℓ
,
j
)
E
n
κ
s
+ (fm−1)
A
=
B
=
0
A
=
0.5
,
B
=
0.2
1
0, −2
0
p
3
/
2
0.882352941
2.408312542
2
0, −3
0
d
5
/
2
0.945945946
2.835633655
3
0, −4
0
f
7
/
2
0.969230769
3.186164523
4
0, −5
0
g
9
/
2
0.980198020
3.494741008
1
1, −2
1
p
3
/
2
0.945945946
2.557131582
2
1, −3
1
d
5
/
2
0.969230769
2.960686798
3
1, −4
1
f
7
/
2
0.980198020
3.301543491
4
1, −5
1
g
9
/
2
0.986206897
3.603991399
1
0, 1
0
p
1
/
2
0.882352941
1.577184026
2
0, 2
0
d
3
/
2
0.945945946
1.665371674
3
0, 3
0
f
5
/
2
0.969230769
1.695353106
4
0, 4
0
g
7
/
2
0.980198020
1.708910609
1
1, 1
1
p
1
/
2
0.945945946
1.798483171
2
1, 2
1
d
3
/
2
0.969230769
1.873601141
3
1, 3
1
f
5
/
2
0.980198020
1.910316883
4
1, 4
1
g
7
/
2
0.986206897
1.931627027Table 3
Energies in the pseudospin symmetry Limit forA=0.5, B=0.2, b=1, M=1 fm−1.
C
ps
E
n
κ
ps
- (fm−1)
1
P
3
/
2
1
f
7
/
2
0
d
3
/
2
0
g
7
/
2
−4
−4.720070702
−5.340430189
−3.401741077
−3.398579611
−2
−3.528809436
−4.224630585
−1.997983133
−1.989883697
0
−2.710867336
−3.373431922
−1.396649362
−1.414060440
2
−2.197514516
−2.788106698
−1.161286022
−1.224585997
4
−1.872532099
−2.315650253
−1.021036465
−1.131350750Table 4
Energies in the pseudospin symmetry limit forA=0.5, B=0.2, b=1, Cps=0fm-1.
M
(
fm
-
1
)
E
n
κ
ps
-
(
fm
-
1
)
1
P
3
/
2
1
f
7
/
2
0
d
3
/
2
0
g
7
/
2
0
−2.528809436
−3.224630585
−0.997983133
−0.989883697
2
−3.197514516
−3.788106698
−2.161286022
−2.224585997
4
−4.653565405
−5.108745157
−3.914496549
−4.070243046
6
−6.373746671
−6.675285047
−5.75746618
−5.983999509Table 5
Energies in the spin symmetry limit forA=0.5, B=0.2, b=1, M=1 fm−1.
C
s
E
n
κ
s
+
(
fm
-
1
)
0
f
7
/
2
1
d
5
/
2
0
P
1
/
2
1
f
5
/
2
−4
2.202940158
1.991277770
0.885667514
1.319214847
−2
2.599395981
2.379236118
1.159816872
1.527861114
0
3.186164523
2.960686798
1.577184026
1.910316883
2
4.044444399
3.826180645
2.362620095
2.672956317
4
5.214113098
5.019180712
3.720450502
3.966069464Table 6
Energies in the spin symmetry limit forA=0.5, B=0.2, b=1, Cs=0fm-1.
M
(
fm
-
1
)
E
n
κ
s
+
(
fm
-
1
)
0
f
7
/
2
1
d
5
/
2
0
P
1
/
2
1
f
5
/
2
0
3.044444399
2.826180645
1.362620095
1.672956317
2
3.599395981
3.379236118
2.159816872
2.527861114
4
4.926530873
4.721303277
3.669041728
4.186965577
6
6.569688910
5.960686798
5.304646313
6.017810527


## 3.1. Solution of the Pseudospin Symmetry Limit

In the previous section, we obtained a Schrödinger-like equation of the form(3.1)d2Gnκps(r)dr2+[εnκps-a1psr2-b1psr+c1psr+d1ps-κ(κ-1)r2]Gnκps(r)=0,
where
(3.2)εnκps=-M2-MCps+(Enκps)2-EnκpsCps+2κA+A-2AB,a1ps=A2,b1ps=-apsM+apsEnκps-apsCps,c1ps=bpsM-bpsEnκps+bpsCps,d1ps=2κB-B-B2.
Equation (3.1) fails to admit exact analytical solutions. Therefore, we follow the ansatz approach with the starting square:
(3.3)Gnκps(r)=fnps(r)exp(gκps(r)),
where
(3.4)fnps(r)={1,ifn=0,∏i=1n(r-αin),ifn≥1,(3.5)gκps(r)=-12αpsr2-βpsr+δpslnr,αps>0,βps>0.
By substitution of fn(r) and gκ(r) into (3.3), we find
(3.6)Gnκps′′(r)=[gκps′′(r)+gκps′2(r)+fnps′′(r)+2gκps′(r)fnps′(r)fnps(r)]Gnκps(r).
Here, we consider the case n=0. From (3.4)–(3.6) we find
(3.7)G0κps′′(r)={(αps)2r2+2αpsβpsr-αps[1+2(δps+0)]+(βps)2-2βps(δps+0)r+δps(δps-1)r2}G0κps(r).
By comparing the corresponding powers of (3.1) and (3.7), we have
(3.8)αps=a1ps,a1ps>0,βps=b1ps2a1ps,c1ps=2βps(δps+0),δps=12(1±(2κ-1)2-4d1ps)=12(1±k1ps),ε0κps=αps[1+2(δps+0)]-(βps)2,
where k1ps=(2κ-1)2-4d1ps. Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take δ from (3.8) as
(3.9)δps=12(1+k1ps).
Form (3.2), (3.8), the ground-state energy satisfies(3.10a)-M2-MCps+(E0κps)2-E0κpsCps+2κA+A-2AB-αps[1+2(δps+0)]+βps2=0
or
(3.10b)(E0κps)2-E0κpsCps-M2-MCps+2κA+A-2AB-a1ps(2+(2κ-1)2-4d1ps)+b1ps24a1ps=0,which is more compactly written as
(3.11)E0κps±=12{+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2Cps±[4κ2-4κ+1-4d1ps-4βps2Cps2-8κA-4A+8AB+4M2+4MCps+8a1ps+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2},(3.35),
where the parameter aps of potential (2.6) from (3.8) should satisfy the following restriction:
(3.12)aps=-2bpsa1ps(1+(2κ-1)2-4d1ps)=-2bpsa1ps(1+k1ps).
From (3.3), (3.4), and (3.8), the upper and lower components of the wave function are(3.13a)gκ(r)=-12a1psr2-b1ps2a1psr+12(1+(2κ-1)2-4d1ps)lnr,(3.13b)G0κps(r)=N0κr(1/2)(1+(2κ-1)2-4d1ps)exp(-12a1psr2-b1ps2a1psr),(3.13c)F0κps(r)=1M-E0κps+Cps(ddr-κr+U(r))G0κps(r).For the first node (n=1), using f1(r)=(r-α1(1)) and gκ(r) from (3.5), we arrive at
(3.14)a1psr2+b1psr-c1psr+κ(κ-1)-d1psr2-ε1κps=-αps-δpsr2+(αps)2r2+(βps)2+(δps)2r2+2αpsβpsr-2δpsαps-2βpsδpsr+-2αpsr-2βps+2δps/rr-α1(1).
Here, the consequent relations between the potential parameters and the coefficients αps, βps, δps, and α1(1) are
(3.15)αps=a1ps,βps=b1ps2a1ps,δps=12(1+k1ps),ε1κps=αps[1+2(δps+1)]-(βps)2,(c1ps-2βpsδps)α1(1)=2δps,c1ps-2βps(δps+1)=2αpsα1(1).
By solving the above equations one can find c1ps and α1(1) as(3.16a)c1ps=b1ps2a1ps((2κ-1)2-4d1ps+2)+b1ps24a1ps+2a1ps(1+(2κ-1)2-4d1ps),αpsα1(1)2+βpsα1(1)-δps=0,(3.16b)→α1(1)=12-βps+(βps)2+4αpsδpsαps=-b1ps4a1ps+b1ps216a1ps2+1+(2κ-1)2-4d1ps2a1ps.The energy eigenvalues therefore are(3.17)E1κps±=12{+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2Cps±[4a1ps4κ2-4κ+1-4d1ps-4βps2Cps2-8κA-4A+8AB+4M2+4MCps+16a1ps+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2},
where the parameter aps from (3.8) is
(3.18)aps=2a1ps(2c1ps+c1psk1ps+c1ps2+6a1ps+14a1psk1ps+10a1psk1ps2+2a1psk1ps3)-3M-4Mk1ps-Mk1ps2+3E1κps+4E1κpsk1ps+E1κpsk1ps2-3Cps-4Cpsk1ps-Cpsk1ps2.
For the upper and lower components of the wave function we thus have(3.19a)G1κps(r)=N1κ(r-α1(1))r(1/2)(1+(2κ-1)2-4d1ps)exp(-12a1psr2-b1ps2a1psr),(3.19b)F1κps(r)=1M-E1κps+Cps(ddr-κr+U(r))G1κps(r).Following the analytic iteration procedures for the second node(n=2) with f2(r)=(r-α1(2))(r-α2(2)) and gκ(r) as defined in (3.6), the relations between the potential parameters and the coefficients αps, βps, δps, α1(2), and α2(2)are
(3.20)αps=a1ps,βps=b1ps2a1ps,δps=12(1+k1ps),ε2κps=αps[1+2(δps+2)]-(βps)2,c1ps-2βps(δps+2)=2αps∑i=12αi(2),(c1ps-2βpsδps)∑i<j2αi(2)αj(2)=2δps∑i=12αi(2),[c1ps-2βps(δps+1)]∑i=12αi(2)=4αps∑i<j2αi(2)αj(2)+2(2δps+1).
The coefficients α1(2) and α2(2) are found from the constraint relations [20, 27–29]:
(3.21)αps∑i=12αi(2)2+βps∑i=12αi(2)-(2δps+1)=0,(3.22)δps∑i=12αi(2)2-(βps∑i=12αi(2)+1)∑j<k2αj(2)αk(2)-2αps∑j<k2αj(2)2αk(2)2=0.
Therefore, the energy eigenvalue in this case is
(3.23)E2κps±=12{+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2Cps±[>+4a1ps4κ2-4κ+1-4d1ps-4βps2Cps2-8κA-4A+8AB+4M2+4MCps+24a1ps+4a1ps4κ2-4κ+1-4d1ps-4βps2]1/2},
and the lower component of the wave function is
(3.24)G2κps(r)=N2κ∏i=12(r-αi(2))r(1/2)(1+(2κ-1)2-4d1ps)exp(-12a1psr2-b1ps2a1psr).


## 3.2. Solution of the Spin Symmetry Limit

In this case, our ordinary differential equation is(3.25)d2Fnκs(r)dr2+[εnκs-a1sr2-b1sr+c1sr+d1s-κ(κ+1)r2]Fnκs(r)=0,
with
(3.26)εnκs=-M2+MCs+(Enκs)2-EnκsCs+2κA-A-2AB,a1s=A2,b1s=asM+asEnκs-asCs,c1s=-bsM-bsEnκs+bsCs,d1s=2κB+B-B2,
which cannot be solved by our common exact analytical techniques. Let us propose the ansatz solution:
(3.27)Fnκs(r)=fns(r)exp(gκs(r)),
where
(3.28)fns(r)={1,ifn=0,∏i=1n(r-αin),ifn≥1,(3.29)gκs(r)=-12αsr2-βsr+δslnr,αs>0,βs>0.
By substitution of fn(r) and gκ(r) into (3.27), we find
(3.30)Fnκs′′(r)=[gκs′′(r)+gκs′2(r)+fns′′(r)+2gκs′(r)fns′(r)fns(r)]Fnκs(r).
For the case of n=0, from (3.26)–(3.29), we find
(3.31)Fnκs′′(r)={(αs)2r2+2αsβsr-αs[1+2(δs+0)]+(βs)2-2βs(δs+0)r+δs(δs-1)r2}Fnκs(r).
By comparing the corresponding powers of (3.27) and (3.33), we have
(3.32)αs=a1s,βs=b1s2a1s,a1s>0,c1s=2βs(δs+0),δs=12(1±(2κ+1)2-4d1s)=12(1±k1s),ε0κs=α[1+2(δs+0)]-(βs)2,
where k1s=(2κ+1)2-4d1s. To have physically acceptable solutions, we pick up the value
(3.33)δs=12(1+k1s).
By considering (3.26), (3.32), the first node eigenvalue satisfies
(3.34)-M2+MCs+(Enκs)2-EnκsCs+2κA-A-2AB-αs[1+2(δs+0)]+(βs)2=0,
or equivalently
(3.35)E0κs±=12{+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs±[+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs2+4M2-4MCs-8κA+4A+8AB+8a1s+4a1s4κ2+4κ+1-4d1s-4βs2+4a1s4κ2+4κ+1-4d1s-4βs2]1/2]1/2},
where the parameter as of potential (2.9a) should satisfy the restriction
(3.36)as=-2bsa1s(1+k1s).
From (3.27), (3.28), and (3.32), the upper and lower components of the wave function are
(3.37)gκ(r)=-12a1sr2-b1s2a1sr+12(1+(2κ+1)2-4d1s)lnr,(3.38)F0κs(r)=N0κr(1/2)(1+(2κ+1)2-4d1s)exp(-12a1sr2-b1s2a1sr),(3.39)G0κs(r)=1M+E0κs-Cs(ddr+κr-U(r))F0κs(r).
Secondly, for the first node (n=1), using f1(r)=(r-α1(1)) and gκ(r) from (3.28)–(3.30), our resulting equation is
(3.40)a1sr2+b1sr-c1sr+-d1s+κ(κ+1)r2-εnκs=-αs-δsr2+(αs)2r2+(βs)2+(δs)2r2+2αsβsr-2δsαs-2βsδsr+-2αsr-2βs+2δs/rr-α1(1).
The relations between the potential parameters and the coefficients αs,βs,δs, and α1(1) are
(3.41)αs=a1s,βs=b1s2a1ps,a1s>0,δs=12(1+(2κ+1)2-4d1s)=12(1+k1s),ε1κs=αs[1+2(δs+1)]-(βs)2,(c1s-2βsδs)α1(1)=2δs,c1s-2βs(δs+1)=2αsα1(1),
where c1s and α1(1) are found from (3.41) as [20, 27–29](3.42a)c1s=b1s2a1s((2κ+1)2-4d1s+2)+b1s24a1s+2a1s(1+(2κ+1)2-4d1s),(3.42b)αsα1(1)2+βsα1(1)-δs=0,
or
(3.42c)α1(1)=12-βs+(βs)2+4αsδsαs=-b1s4a1s+b1s216a1s2+1+(2κ+1)2-4d1s2a1s,which determine the corresponding energy as
(3.43)E1κs±=12{+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs±[4κ2+4κ+1-4d1s-4βs2Cs2+4M2-4MCs-8κA+4A+8AB+16a1s+4a1s4κ2+4κ+1-4d1s-4βs2]1/2},
with
(3.44)as=2a1s(2c1s+c1sk1s+c1s2+6a1s+14a1sk1s+10a1sk1s2+2a1sk1s3)3M+4Mk1s+Mk1s2+3E1κs+4E1κsk1s+E1κsk1s2-3Cs-4Csk1s-Csk1s2.
The upper and lower components of the wave function are then simply found to be(3.45a)F1κs(r)=N1κ(r-α1(1))r(1/2)(1+(2κ+1)2-4d1s)exp(-12a1sr2-b1s2a1sr),(3.45b)G1κs(r)=1M+E1κs-Cs(ddr+κr-U(r))F1κs(r).For the second node(n=2), we choose f2(r)=(r-α1(2))(r-α2(2)) and gκ(r) as defined in (3.28) and (3.29). The relations between the potential parameters and the coefficients αs,βs,δs,α1(2), and α2(2) are
(3.46)αs=a1s,βs=b1s2a1s,δs=12(1+k1s),ε2κs=αs[1+2(δs+2)]-(βs)2,c1s-2βs(δs+2)=2αs∑i=12αi(2),(c1s-2βsδs)∑i<j2αi(2)αj(2)=2δs∑i=12αi(2),[c1s-2βs(δs+1)]∑i=12αi(2)=4α∑i<j2αi(2)αj(2)+2(2δs+1),
where
(3.47)αs∑i=12αi(2)2+βs∑i=12αi(2)-(2δs+1)=0,(3.48)δs∑i=12αi(2)2-(βs∑i=12αi(2)+1)∑j<k2αj(2)αk(2)-2αs∑j<k2αj(2)2αk(2)2=0.
The energy eigenvalue therefore is
(3.49)E2κs±=12{+4a1s4κ2+4κ+1-4d1s-4βs2]1/2Cs±[a1s4κ2+4κ+1-4d1s-4βs2Cs2+4M2-4MCs-8κA+4A+8AB+24a1s+4a1s4κ2+4κ+1-4d1s-4βs2]1/2}.
For the upper component of the wave function, we have
(3.50)F2κs(r)=N2κ∏i=12(r-αi(2))r(1/2)(1+(2κ+1)2-4d1s)exp(-12a1sr2-b1s2a1sr).
We have given some numerical values of the energy eigenvalues in Tables 1, 2, 3, 4, 5, and 6 for various states. For the final point, we wish to emphasize on the degeneracy-removing role of the (Cornell) tensor potential. As we already know, for vanishing tensor interaction (A=B=0), the pseudospin doublets, that is, states with quantum numbers (n,l,j=l+1/2) and (n-1,l+2,j=l+3/2) are degenerate. The degenerate states in the spin doublets are those with quantum numbers (n,ℓ,j=ℓ+1/2) and (n,ℓ,j=ℓ-1/2), where n,l,and j are the radial, the orbital, and the total angular momentum quantum numbers, respectively (see Tables 1 and 2). Our numerical data reveals that, in the pseudospin symmetry limit, the degenerate states for (A=B=0) are ns1/2,(n-1)d3/2 for ℓ~=1(ℓ=0), np3/2,(n-1)f5/2 for ℓ~=2(ℓ=1), nd5/2,(n-1)g7/2 for ℓ~=3(ℓ=2), nf7/2,(n-1)h9/2 for ℓ~=4(ℓ=3), and so forth. For spin symmetry limit and (A=B=0) one can clearly see that the degeneracy occurs in (np1/2,np3/2) for ℓ=1, (nd3/2,nd5/2) for ℓ=2,(nf5/2,nf7/2) for ℓ=3, (ng7/2,ng9/2) for ℓ=4, and so forth.Table 1
Bound state for the pseudospin symmetryb=1, M=1 fm−1, Cps=0 fm−1.
ℓ
~
n
,
κ
(
ℓ
,
j
)
E
n
κ
ps
- (fm−1)
A
=
B
=
0
A
=
0.5
,
B
=
0.2
1
0, −1
0
S
1
/
2
−0.882352941
−1.984891569
2
0, −2
0
P
3
/
2
−0.945945946
−2.471004354
3
0, −3
0
d
5
/
2
−0.969230769
−2.861584264
4
0, −4
0
f
7
/
2
−0.980198020
−3.199522676
1
1, −1
1
S
1
/
2
−0.945945946
−2.302587881
2
1, −2
1
p
3
/
2
−0.969230769
−2.710867336
3
1, −3
1
d
5
/
2
−0.980198020
−3.060956677
4
1, −4
1
f
7
/
2
−0.986206897
−3.373431922
1
1, 2
0
d
3
/
2
−0.945945946
−1.396649362
2
1, 3
0
f
5
/
2
−0.969230769
−1.411440323
3
1, 4
0
g
7
/
2
−0.980198020
−1.414060440
4
1, 5
0
h
9
/
2
−0.986206897
−1.414104092Table 2
Bound state for the spin symmetryb=1, M=1 fm−1, Cs=0 fm−1.
ℓ
n
,
κ
(
ℓ
,
j
)
E
n
κ
s
+ (fm−1)
A
=
B
=
0
A
=
0.5
,
B
=
0.2
1
0, −2
0
p
3
/
2
0.882352941
2.408312542
2
0, −3
0
d
5
/
2
0.945945946
2.835633655
3
0, −4
0
f
7
/
2
0.969230769
3.186164523
4
0, −5
0
g
9
/
2
0.980198020
3.494741008
1
1, −2
1
p
3
/
2
0.945945946
2.557131582
2
1, −3
1
d
5
/
2
0.969230769
2.960686798
3
1, −4
1
f
7
/
2
0.980198020
3.301543491
4
1, −5
1
g
9
/
2
0.986206897
3.603991399
1
0, 1
0
p
1
/
2
0.882352941
1.577184026
2
0, 2
0
d
3
/
2
0.945945946
1.665371674
3
0, 3
0
f
5
/
2
0.969230769
1.695353106
4
0, 4
0
g
7
/
2
0.980198020
1.708910609
1
1, 1
1
p
1
/
2
0.945945946
1.798483171
2
1, 2
1
d
3
/
2
0.969230769
1.873601141
3
1, 3
1
f
5
/
2
0.980198020
1.910316883
4
1, 4
1
g
7
/
2
0.986206897
1.931627027Table 3
Energies in the pseudospin symmetry Limit forA=0.5, B=0.2, b=1, M=1 fm−1.
C
ps
E
n
κ
ps
- (fm−1)
1
P
3
/
2
1
f
7
/
2
0
d
3
/
2
0
g
7
/
2
−4
−4.720070702
−5.340430189
−3.401741077
−3.398579611
−2
−3.528809436
−4.224630585
−1.997983133
−1.989883697
0
−2.710867336
−3.373431922
−1.396649362
−1.414060440
2
−2.197514516
−2.788106698
−1.161286022
−1.224585997
4
−1.872532099
−2.315650253
−1.021036465
−1.131350750Table 4
Energies in the pseudospin symmetry limit forA=0.5, B=0.2, b=1, Cps=0fm-1.
M
(
fm
-
1
)
E
n
κ
ps
-
(
fm
-
1
)
1
P
3
/
2
1
f
7
/
2
0
d
3
/
2
0
g
7
/
2
0
−2.528809436
−3.224630585
−0.997983133
−0.989883697
2
−3.197514516
−3.788106698
−2.161286022
−2.224585997
4
−4.653565405
−5.108745157
−3.914496549
−4.070243046
6
−6.373746671
−6.675285047
−5.75746618
−5.983999509Table 5
Energies in the spin symmetry limit forA=0.5, B=0.2, b=1, M=1 fm−1.
C
s
E
n
κ
s
+
(
fm
-
1
)
0
f
7
/
2
1
d
5
/
2
0
P
1
/
2
1
f
5
/
2
−4
2.202940158
1.991277770
0.885667514
1.319214847
−2
2.599395981
2.379236118
1.159816872
1.527861114
0
3.186164523
2.960686798
1.577184026
1.910316883
2
4.044444399
3.826180645
2.362620095
2.672956317
4
5.214113098
5.019180712
3.720450502
3.966069464Table 6
Energies in the spin symmetry limit forA=0.5, B=0.2, b=1, Cs=0fm-1.
M
(
fm
-
1
)
E
n
κ
s
+
(
fm
-
1
)
0
f
7
/
2
1
d
5
/
2
0
P
1
/
2
1
f
5
/
2
0
3.044444399
2.826180645
1.362620095
1.672956317
2
3.599395981
3.379236118
2.159816872
2.527861114
4
4.926530873
4.721303277
3.669041728
4.186965577
6
6.569688910
5.960686798
5.304646313
6.017810527


## 4. Conclusion

Because of the established roles of the Cornell potential and spin, and pseudospin symmetries in nuclear and hadrons spectroscopy, we solved the Dirac equation under these symmetry limits for vector, scalar, and tensor interactions of Cornell-type. In our calculations, on the one hand, due to the failure of other common analytical techniques, and, on the other hand, the better insight which analytical techniques provide us in comparison with their numerical counterparts, we used the quasianalytical ansatz approach. By proposing novel physical solutions and after lengthy calculations, we could find the arbitrary-state solutions. Our results clearly show the degeneracy-removing role of the tensor term and provide the requisite understanding of the solutions for possible further studies. Both the energy spectrum and the eigenfunctions can be used in related systems after the proper phenomenological fits done.



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*Source: 707041-2012-10-03.xml*