Acta Polytechnica https://doi.org/10.14311/AP.2023.63.0132 Acta Polytechnica 63(2):132–139, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague EXACT SOLUTIONS FOR TIME-DEPENDENT COMPLEX SYMMETRIC POTENTIAL WELL Boubakeur Khantoula, b, Abdelhafid Bounamesa, ∗ a University of Jijel, Department of Physics, Laboratory of Theoretical Physics, BP 98 Ouled Aissa, 18000 Jijel, Algeria b University of Constantine 3 – Salah Boubnider University, Department of Process Engineering, BP B72 Ali Mendjeli, 25000 Constantine, Algeria ∗ corresponding author: bounames@univ-jijel.dz Abstract. Using the pseudo-invariant operator method, we investigate the model of a particle with a time-dependent mass in a complex time-dependent symmetric potential well V (x,t) = if (t) |x|. The problem is exactly solvable and the analytic expressions of the Schrödinger wavefunctions are given in terms of the Airy function. Indeed, with an appropriate choice of the time-dependent metric operators and the unitary transformations, for each region, the two corresponding pseudo-Hermitian invariants transform into a well-known time-independent Hermitian invariant which is the Hamiltonian of a particle confined in a symmetric linear potential well. The eigenfunctions of the last invariant are the Airy functions. Then, the phases obtained are real for both regions and the general solution to the problem is deduced. Keywords: Non-Hermitian Hamiltonian, time-dependent Hamiltonian, pseudo-invariant method, PT-symmetry, pseudo-Hermiticity. 1. Introduction The discovery of a class of non-Hermitian Hamilto- nian that may have a real spectrum has prompted a revival of theoretical and applied research in quan- tum physics. In fact, in 1998, C.M. Bender and S. Boettcher showed that any non-Hermitian Hamilto- nian invariant under the unbroken space-time reflec- tion, or PT -symmetry, has real eigenvalues and satis- fies all the physical axioms of quantum mechanics [1–3]. In 2002, A. Mostafazadeh presented a more extended version of non-Hermitian Hamiltonians having a real spectrum, proving that the hermiticity of the Hamilto- nian with respect to a positive definite inner product, ⟨., .⟩η = ⟨.| η |.⟩, is a necessary and sufficient condition for the reality of the spectrum, where η is the met- ric operator which is linear, Hermitian, invertible and positive. This condition requires that the Hamiltonian H satisfies the pseudo-Hermitian relation [4–6]: H† = ηHη† . (1) Moreover in recent years, a significant progress has been achieved in the study of time-dependent (TD) non-Hermitian quantum systems in several branches of physics. Finding exact solutions to the TD Schrödinger equation, which cannot be reduced to eigenvalues equation in general, is a problem of intriguing difficulty. Different methods are used to ob- tain solutions of Schrödinger’s equation for explicitly TD systems, such as unitary and non-unitary transfor- mations, the pseudo-invariant method, Dyson’s maps, point transformations, Darboux transformations, per- turbation theory and adiabatic approximation [7–31]. However, the emergence of a non-linear Ermakov-type auxiliary equation for several TD systems, which is dif- ficult to solve, constitutes an additional constraint to obtain exact analytical solutions [32, 33]. This greatly reduces the number of exactly solvable time-dependent non-Hermitian systems [34–38]. In particular, other works have been concerned with studying exact so- lutions of TD Hamiltonians with a specific TD mass in the non-Hermitian case [39, 40] and also in the Hermitian case [41–45]. In the present work, we used the pseudo-invariant method [17] to obtain the exact solutions of the Schrödinger equation for a particle with TD mass moving in a TD complex symmetric potential well: V (x,t) = if(t) |x| , (2) where f(t) is an arbitrary real TD function. The manuscript is organised as follows: In Sec- tion 2, we introduce some of the basic equations of the TD non-Hermitian Hamiltonians and their time-dependent Schrödinger equation (TDSE) with a TD metric. In Section 3, we discuss the use of the Lewis-Riesenfeld invariant method to address the Schrödinger equation for an explicitly TD non- Hermitian Hamiltonian. In Section 4, we use the Lewis-Riesenfeld method to solve the TD Schrödinger equation for a particle with TD mass in a TD complex symmetric potential well. Finally, in Section 5, we conclude with a brief review of the obtained results. 132 https://doi.org/10.14311/AP.2023.63.0132 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 63 no. 2/2023 Exact solutions for time-dependent complex symmetric potential well 2. TD Non-Hermitian Hamiltonian with TD metric Let H(t) be a non-Hermitian TD Hamiltonian and h(t) its associated TD Hermitian Hamiltonian. The two corresponding TD Schrödinger equations describing the quantum evolution are: H(t) ∣∣ΦH (t)〉 = iℏ∂t ∣∣ΦH (t)〉 , (3) h(t) ∣∣Ψh(t)〉 = iℏ∂t ∣∣Ψh(t)〉 , (4) where the two Hamiltonians are related by the Dyson maps ρ(t) as H(t) = ρ−1(t) h(t) ρ(t) − iℏρ−1(t) ρ̇(t) , (5) and their wavefunctions ∣∣ΦH (t)〉 and ∣∣Ψh(t)〉 as∣∣Ψh(t)〉 = ρ(t) ∣∣ΦH (t)〉 . (6) The hermiticity of h(t) allowed us to establish the connection between the Hamiltonian H(t) and its Hermitian conjugate H†(t) as H†(t) = η(t) H(t) η−1(t) + iℏη̇(t) η−1(t) , (7) which is a generalisation of the well-known conven- tional quasi-Hermiticity Equation (1), and the TD metric operator is Hermitian and defined as η(t) = ρ†(t) ρ(t). 3. Pseudo-invariant operator method Let us start with the description of the Lewis- Riesenfeld theory [46] for a TD Hermitian Hamil- tonian h(t) with a Hermitian TD invariant Ih(t). The dynamic invariant Ih(t) satisfies: dIh(t) dt = ∂Ih(t) ∂t − i ℏ [ Ih(t) , h(t) ] = 0 . (8) The eigenvalue equation for Ih(t) is: Ih(t) ∣∣ψhn(t)〉 = λn ∣∣ψhn(t)〉 , (9) where the eigenvalues λn of Ih(t) are reals and time- independent, and the Lewis-Riesenfeld phase is defined as: ℏ d dt εn(t) = 〈 ψhn(t) ∣∣ iℏ ∂ ∂t − h(t) ∣∣ψhn(t)〉 , (10) and the solution of the TDSE of h(t) is given as∣∣Ψh(t)〉 = exp [iεn(t)] ∣∣ψhn(t)〉 . (11) In the paper [17], we showed that any TD Hamilto- nian H(t) satisfying the TD quasi-hermiticity Equa- tion (7) admits a pseudo-hermitician invariant Iph(t) such that: Iph†(t) = η(t)Iph(t)η−1(t) ⇔ Ih(t) = ρ(t)Iph(t)ρ−1(t) = Ih†(t) . (12) Since the Hermitian invariant Ih(t) satisfies the eigenvalues Equation (9), Equation (12) ensures that the pseudo-Hermitian invariant’s spectrum is real with the same eigenvalues λn of Ih(t): Ih(t) ∣∣ψhn(t)〉 = λn ∣∣ψhn(t)〉 , (13) Iph(t) ∣∣ϕphn (t)n(t)〉 = λn ∣∣ϕphn (t)〉 , (14) where the eigenfunctions ∣∣ψhn(t)〉 and ∣∣ϕphn (t)〉, of Ih(t) and Iph(t), respectively, are related as∣∣ψhn(t)〉 = ρ(t) ∣∣ϕphn (t)〉 . (15) The inner products of the eigenfunctions associated with the non-Hermitian invariant Iph(t) can now be written as ⟨ϕphm (t) ∣∣ϕphn (t)〉η = ⟨ϕphm (t)|η ∣∣ϕphn (t)〉 = δmn , (16) and it corresponds to the conventional inner product associated to the Hermitian invariant Ih(t). It is easy to verify, by a direct substitution of the Hermitian Hamiltonian h(t) and the Hermitian invari- ant Ih(t) by their equivalents in the Expressions (5) and (12), respectively, that the pseudo Hermitian invariant Iph(t) satisfies: ∂Iph(t) ∂t = i ℏ [ Iph(t) , H(t) ] . (17) We should remark that the invariant operator’s eigenstates and eigenvalues can be computed using the same procedure as the Hermitian case. The solution ∣∣ΦH (t)〉 of the Schrödinger Equa- tion (3) is different from ∣∣ϕphn (t)〉 in Equation (14) only by the factor eiε ph n (t) where εphn (t) is a real phase given by: ℏ d dt ε ph n (t) = 〈 ϕ ph n (t) ∣∣ η(t) [iℏ ∂ ∂t − H(t) ] ∣∣ϕphn (t)〉 . (18) 4. Particle in TD complex symmetric potential well Let us consider a particle with a TD mass m(t) in the presence of a pure imaginary TD symmetric poten- tial well Equation (2), where its Hamiltonian can be written as: H(t) = { p2 2m(t) + if(t)x if x ≥ 0 p2 2m(t) − if(t)x if x ≤ 0 , (19) the associated TDSE of the system is: 133 Boubakeur Khantoul, Abdelhafid Bounames Acta Polytechnica [ p2 2m(t) + if(t) |x| ] Ψ(x,t) = i ∂ ∂t Ψ(x,t) , (20) where m(t) is the particle TD mass and f(t) an arbi- trary real TD function, and the unit of ℏ = 1. This model can be considered as the complex version of the Hermitian case of a particle, with TD mass and charge q, moving under the action of TD electric field E(t) and confined in a pure imaginary symmetric linear potential well: if(t)x for x ≥ 0 and −if(t)x for x ≤ 0, where f(t) = −qE(t). According to the results in [17], the solution to the TD Schrödinger equation with a TD non-Hermitian Hamiltonian is easily found if a nontrivial TD pseudo- Hermitian invariant Iph(t) exists and satisfies the von- Neumann Equation (17). In the current problem, in order to solve the TD Shrödinger Equation (20) we assume that the Hamil- tonian H(t) admits an invariant in each region: let I ph 1 (t) for x ≥ 0 and I ph 2 (t) for x ≤ 0. For the region x ≥ 0, let us look for a non-Hermitian TD invariant in the following quadratic form: I ph 1 (t) = β1(t)p 2 + β2(t)x + β3(t)p + β4(t) , (21) where βi(t) are arbitrary complex functions to be determined. By inserting the Expressions (19) and (21) in Equation 17, the following system of equations can be found:   β̇1(t) = 0 , β̇2(t) = 0 , β̇3(t) = − β2(t) m(t) + 2 if(t)β1(t) , β̇4(t) = if(t)β3(t) , (22) to simplify the calculations, we take β1(t) = 1 and β2(t) = 1, so β3(t) and β4(t) are given by: β3(t) = g(t) + ik(t) , (23) β4(t) = s(t) + iw(t) , (24) where g(t) = − ∫ dt m(t) , k(t) = 2 ∫ f(t)dt, s(t) = − ∫ f(t)k(t)dt and w(t) = ∫ f(t)g(t)dt. Substituting Expressions (23) and (24) in Equa- tion (21) we found: I ph 1 (t) = p 2 + x + [g(t) + ik(t)] p + s(t) + iw(t) . (25) Its eigenvalue equation is as follows: I ph 1 (t) |ψ(t)⟩ = λ1 |ψ(t)⟩ , (26) in order to show that the spectrum of Iph1 (t) is real, we search for a metric operator that fulfills the pseudo hermiticity relation: I ph† 1 (t) = η1(t)I ph 1 (t)η −1 1 (t) , (27) and we make the following choice for metric: η1(t) = exp[−α(t)x − β(t)p] , (28) where α(t) and β(t) are chosen as real functions in order that the metric operator η1(t) is Hermitian. The position and momentum operators transform according to the transformation η1(t) as: η1(t)xη−11 (t) = x + iβ(t) , (29) η1(t)pη−11 (t) = p − iα(t) , (30) incorporating these relationships into Equation (27), we found: α(t) = k(t) , (31) β(t) = g(t)k(t) − 2w(t) , (32) then the TD metric operator η1(t) is given by: η1(t) = exp[−k(t)x − (g(t)k(t) − 2w(t))p] , (33) according to the relation η1(t) = ρ † 1(t)ρ1(t), and since ρ1(t) is not unique, we can take it as a Hermitian operator in order to simplify the calculations: ρ1(t) = exp [ − k(t) 2 x − [ g(t)k(t) 2 − w(t) ] p ] , (34) the Hermitian invariant Ih1 (t) associated with the pseudo-Hermitian invariant Iph1 (t) is given by: I h 1 (t) = ρ(t)I ph 1 ρ −1(t) = p2 +x+g(t)p+ k2(t) 4 +s(t) . (35) For the region x ≤ 0, we take the non-Hermitian invariant Iph2 as I ph 2 (t) = α1(t)p 2 + α2(t)x + α3(t)p + α4(t) , (36) where αi(t) are arbitrary complex functions to be determined. In the same way as the precedent case, inserting the Expressions (19) and (36) in Equation (17), where we take α1(t) = 1 and α2(t) = −1, so α3(t) and α4(t) are given by: α3(t) = −g(t) − ik(t) , (37) α4(t) = s(t) + iw(t) . (38) 134 vol. 63 no. 2/2023 Exact solutions for time-dependent complex symmetric potential well Then, the final results of Iph2 (t) and η2(t) are: I ph 2 (t) = p 2 − x − [g(t) + ik(t)] p + s(t) + iw(t) , (39) η2(t) = exp[k(t)x − [2w(t) − g(t)k(t)] p] . (40) We take ρ2(t) as a Hermitian operator, then η2(t) = ρ22, ρ2(t) = exp [ k(t) 2 x + [ k(t)g(t) 2 − w(t) ] p ] , (41) and the related Hermitian invariant Ih2 (t) is: Ih2 (t) = p 2 − x − g(t)p + k2(t) 4 + s(t) . (42) To derive the eigenvalues equations of the invariants Ihj (t) for the two regions (j = 1, 2), we introduce the unitary transformations Uj (t): |ϕn,j (t)⟩ = Uj (t) |φn⟩ , j = 1, 2 , (43) where φn will be determined later and U1(t) = exp [ −i g(t) 2 x + i 4 [ k2(t) − g2(t) + 4s(t) ] p ] , (44) U2(t) = exp [ i g(t) 2 x − i 4 [ k2(t) − g2(t) + 4s(t) ] p ] . (45) According to these transformations, the invariants Ih1 (t) and Ih2 (t) turn into: I1 = U † 1 (t)I h 1 (t)U1(t) = p 2 + x, (46) I2 = U † 2 (t)I h 2 (t)U2(t) = p 2 − x, (47) and they can be written in the following combined form: I = p2 + |x| . (48) We note here that I can be considered as the Hamil- tonian of a particle of mass m0 = 1/2 confined in the linear symmetric potential well |x|. Therefore, the eigenvalue equation of the invariant I:[ d2 dx2 + (λn − |x|) ] φn(x) = 0 , (49) is a well-known problem in quantum mechanics. The bound states φn(x) are given in terms of the Airy functions Ai and Bi [47, 48]: φn(x) = Nn Ai(|x| − λn) + N ′ n Bi(|x| − λn) . (50) This solution is not relevant because Bi(|x|−λn) tends to infinity for (|x| − λn) > 0. Thus, we take N ′ n = 0 and the above solution reduces to: φn(x) = Nn Ai(|x| − λn) . (51) The eingenvalues λn are determined by matching the functions φn(x) and their derivatives in the two regions at the point x = 0: φ(1)n (0) = φ (2) n (0) , (52) φ ′(1) n (0) = ±φ ′(2) n (0) , (53) from which there are two possibilities for λn and the normalisation constant Nn depending on whether n is even or odd: • If n is even: λn = −a′n 2 +1 , (54) where a′k is the k th zero of the derivative Ai′ of the Airy function, and all values of a′k are negative numbers [49]. The normalisation constant is: Nn = 1√ −2a′n 2 +1 Ai(a′n 2 +1 ) , (55) and the corresponding eigenfunction of I is: φn(x) = 1√ −2a′n 2 +1 Ai(a′n 2 +1 ) Ai(|x| + a′n 2 +1 ) , (56) • If n is odd: λn = −a n+1 2 , (57) where ak is the kth zero of the Airy function Ai, and all values of ak are negative numbers [49]. The normalisation constant is: Nn = 1 √ 2Ai′(a n+1 2 ) , (58) and the corresponding eigenfunction of I is: φn(x) = sgn(x) 1 √ 2Ai′(a n+1 2 ) Ai(|x| +a n+1 2 ) . (59) The eigenfunctions of the Hermitian invariants Ihj (t) are written for each region as: |ϕn,j (t)⟩ = Uj (t) |φn⟩ , (60) then, the eigenfunctions of the pseudo-Hermitian in- variants Iphj (t) are given by: |ψn,j (t)⟩ = ρ−1j (t)Uj (t) |φn⟩ , (61) 135 Boubakeur Khantoul, Abdelhafid Bounames Acta Polytechnica thus, the solutions of the time-dependent Schrödinger Equation (20) take the form: |Ψn,j (t)⟩ = eiϵ j n(t) |ψn,j (t)⟩ , (62) where ϵjn(t) is the phase (ϵ1n(t) for x ≥ 0 and ϵ2n(t) for x ≤ 0), which is obtained from the following relation: ϵ̇jn(t) = ⟨ψn,j (t)| ηj (t) [ i ∂ ∂t − H(t) ] |ψn,j (t)⟩ = ⟨ϕn,j (t)| iρj (t)ρ̇−1j (t) |ϕn,j (t)⟩ − ⟨ϕn,j (t)| ρj (t)H(t)ρ−1j (t) |ϕn,j (t)⟩ + ⟨ϕn,j (t)| i ∂ ∂t |ϕn,j (t)⟩ = θ(t) − ⟨ϕn,j (t)| p2 2m(t) |ϕn,j (t)⟩ + ⟨ϕn,j (t)| i ∂ ∂t |ϕn,j (t)⟩ , (63) where θ(t) = 1 2 f(t) [ k(t) 2 g(t) − w(t) ] . (64) Using the unitary transformations Uj (t), we found: ϵ̇jn(t) = χ j (t) − 1 2m(t) ⟨φn(t)| (p2 ± x) |φn(t)⟩ , (65) where χ1(t) = θ(t)− 1 16m(t) [ k2(t) + 3g2(t) + 4s(t) ] , (66) χ2(t) = θ(t) + 1 16m(t) [ k2(t) − g2(t) + 4s(t) ] . (67) From the eigenvalue equation of the invariant I, we have: (p2 ± x) |φn(t)⟩ = λn |φn(t)⟩ , (68) then, the phases ϵjn(t) take the form: ϵjn(t) = ∫ (χj (t) − λn 2m(t) )dt, (69) and the solution of the TD Schrödinger Equation (20) is given by: |Ψn,j (t)⟩ = exp [ iϵjn(t) ] ρj (t)−1 |ϕn,j (t)⟩ . (70) In position representation we have: 〈 x ∣∣ρ−1j (t)∣∣ ϕj (t)〉 = exp [iζ(t)] exp [ ± k(t) 2 x ] × ϕj (x ± i( g(t)k(t) 2 − w(t)), t) , (71) where (+) is for the positive region while (−) is for the negative region, and ζ(t) = − k 4 ( g(t)k(t) 2 − w(t)) . (72) Then, the solution of the Schrödinger Equation for each region (70) can be written as: Ψn,j (x,t) = exp [ i(ϵjn(t) + ζ(t)) ] exp [ ± k(t) 2 x ] × ϕn,j (x ± i( g(t)k(t) 2 − w(t)), t) , (73) and the general solution of the Schrödinger Equa- tion (20) is given by: Ψ(x,t) = { Ψn,1(x,t) for x ≥ 0 , Ψn,2(x,t) for x ≤ 0 . (74) According to the Equations (51), (60), (61) and (62), the probability density function is given by: |ρ1(t)Ψn,1| 2 + |ρ2(t)Ψn,2| 2 = = |ϕn,1| 2 + |ϕn,2| 2 = |φn| 2 , (75) and because φn(x) is determined in terms of Airy function Ai(x), which is a real function, and according to Equations (56) and (59), the probability density expression can be written as: • For n is even |φn(x)| 2 = 1 (−2a′n 2 +1 ) [ Ai(a n 2 +1) ]2 [Ai(|x| + a′n2 +1)]2 , (76) and which is represented in Figure 1 for the first three even states (n = 0, 2, 4). • For n is odd |φn(x)| 2 = 1 2 [ Ai′(a n+1 2 ) ]2 [Ai(|x| + a n+12 )]2 , (77) and which is represented in Figure 2 for the first three odd states (n = 1, 3, 5). We note here that the probability in the region x ≤ 0 is ⟨Ψn,2(t)| η2(t) |Ψn,2(t)⟩ = ⟨φn| φn⟩x≤0 = 0∫ −∞ φ∗n(x)φn(x)dx = 1 2 , (78) and the probability in the region x ≥ 0 is 136 vol. 63 no. 2/2023 Exact solutions for time-dependent complex symmetric potential well -6 -4 -2 2 4 6 x 0.1 0.2 0.3 0.4 0.5 φn 2 n=4 n=2 n=0 Figure 1. Probability density of Equation 76 for even values of n = 0, 2, 4. ⟨Ψn,1(t)| η1(t) |Ψn,1(t)⟩ = ⟨φn| φn⟩x≥0 = ∞∫ 0 φ∗n(x)φn(x)dx = 1 2 . (79) So the two regions are equiprobable and the proba- bility in all space is equal to one ⟨Ψ(t) , Ψ(t)⟩η = ⟨Ψn,1| η1(t) |Ψn,1⟩ + ⟨Ψn,2| ηn,2(t) |Ψ2⟩ = ∞∫ −∞ φ∗n(x)φn(x)dx = 1 . (80) 5. Conclusion The pseudo-invariant method has been used to obtain the exact analytical solutions of the time- dependent Schrödinger equation for a particle with time-dependent mass moving in a complex time- dependent symmetric potential well. We have shown that the problem can be reduced to solve a well-known eigenvalue equation for a time-independent Hermitian invariant. In fact, with a specific choice of the TD metric operators, η1(t) and η2(t), and the Dyson maps, ρ1(t) and ρ2(t), and using unitary transformations, the pseudo-invariants operators (Iph1 (t) for x ⩾ 0 and I ph 2 (t) for x ⩽ 0) are mapped to two time-independent Hermitian invariants Ih1 (t) and Ih2 (t), which can be combined in a unique form I = p2 + |x|. The latter can be considered as the Hamiltonian of a particle con- fined in a linear time-independent symmetric potential well, where its eigenfunctions are given in terms of the Airy function Ai. The phases have been calculated for the two regions and are real. Thus, the exact analytical solution of the problem has been deduced. Finally, let us highlight the fact that the probabil- ity density associated with the model in question is time-independent. -6 -4 -2 2 4 6 x 0.05 0.10 0.15 0.20 0.25 0.30 φn 2 n=5 n=3 n=1 Figure 2. Probability density of Equation 77. for odd values of n = 1, 3, 5. References [1] C. Bender, S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters 80(24):5243–5246, 1998. https://doi.org/10.1103/PhysRevLett.80.5243 [2] F. Bagarello, J. Gazeau, F. Szaraniec, M. Znojil. Non-Self adjoint Operators in Quantum Physics: Mathematical Aspects. John Wiley, 2015. [3] C. Bender. PT symmetry: In quantum and classical physics. World Scientific, 2019. [4] A. Mostafazadeh. Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. Journal of Mathematical Physics 43(1):205–214, 2002. https://doi.org/10.1063/1.1418246 [5] A. Mostafazadeh. 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Cambridge University Press, 2010. http://dlmf.nist.gov/9 139 https://doi.org/10.1088/1402-4896/ac3dbd http://arxiv.org/abs/2205.05741 https://doi.org/10.1007/BF02960144 https://doi.org/10.1143/ptp/3.4.440 https://doi.org/10.1103/PhysRevA.33.2870 https://doi.org/10.1142/S0217984918502354 https://doi.org/10.14311/AP.2022.62.0211 https://doi.org/10.1063/1.1664991 http://dlmf.nist.gov/9 Acta Polytechnica 63(2):132–138, 2023 1 Introduction 2 TD Non-Hermitian Hamiltonian with TD metric 3 Pseudo-invariant operator method 4 Particle in TD complex symmetric potential well 5 Conclusion References