Exact solutions of a quartic potential

Qian, Dong; Sun, Guo Hua; Aoki, Manuel Avila; Chen, Chang Yuan

doi:10.1142/s0217732319502080

Cited by 31 publications

(24 citation statements)

References 44 publications

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“…In GL theory, αfalse~→ Tfalse→Tc 0. The Sombrero potential is discussed in several quantum studies due to its relevance in nonlinear quantum dynamics [70–72]. In GL theory, the magnetic vector potential A and consequently the magnetic field boldB=normal∇×boldA play a leading role as emphasized in London's theory.…”

Section: Fractal Gl Theory Of Superconductivitymentioning

confidence: 99%

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

El-Nabulsi

2021

Proc. R. Soc. A.

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Superconductivity is analysed based on the product-like fractal measure approach with fractal dimension α introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media. Our study shows the emergence of a massless state at the boundary of the superconductor and the simultaneous occurrence of isothermal and adiabatic processes in the superconductor depending on the position of the electrons. Several physical quantities were found to be position-dependent comparable with those arising in heavy doping and p–n junction. At the boundary of the superconductor, a shrinkage of the magnetic field was observed, leading to a scenario equivalent to the Meissner–Ochsenfeld effect. An enhancement of the London penetration depth is revealed and such an improvement was observed in pnictides at the onset of commensurate spin-density-wave order inside the superconducting phase at zero temperature. The Bardeen–Cooper–Schrieffer theory was also analysed and the appearance of zero-energy states is detected. Nucleation of superconductivity in a bulk was also studied. The system acts as a quantum damped harmonic oscillator and our analysis showed that type-I superconductivity occurs when κ < 2 / ( 1 + α ) , whereas type II occurs for κ > 2 / ( 1 + α ) , where κ is the Ginzburg–Landau parameter. The transition at the passage from the ‘genuine’ to the ‘intermediate’ type-I estimates 0.767767 < α ≤ 1 .

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Section: Fractal Gl Theory Of Superconductivitymentioning

confidence: 99%

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

El-Nabulsi

2021

Proc. R. Soc. A.

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“…We can separate variables in the Schrödinger equation using the symmetries of (17). The potentials A θ , A z and V (r) are independent of z and θ so that Ĥ, pz = −i ∂ z and pθ = −i ∂ θ can be chosen to have common eigenstates.…”

Section: Quantization Of the Electromagnetic Hamiltonianmentioning

confidence: 99%

“…Quantum anharmonic oscillators have been studied in various contexts and several results are available. For instance, the Schrödinger eigenvalue problem for the one-dimensional (1D) quartic oscillator may be reduced [17] to the triconfluent Heun equation ( [0, 0, 1 6 ] in Ince's classification, see Appendix A). The energy levels of this oscillator display remarkable analytic properties in the complex coupling constant plane [18].…”

Section: Quantum Rajeev-ranken Modelmentioning

confidence: 99%

Quantum Rajeev-Ranken model as an anharmonic oscillator

Krishnaswami,

Vishnu

2021

Preprint

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The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 1 6 ] and lies beyond the ellipsoidal Lamé and Heun equations in Ince's classification. At strong coupling λ , the energies of highly excited states are shown to depend on the scaling variable λk . The energy spectrum at weak coupling and its dependence on wavenumber k in a double-scaling strong coupling limit are obtained. The semi-classical WKB quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk) 2/3 dispersion relation for highly energetic quantized 'screwons' at moderate and strong coupling. We also suggest a mapping between our radial equation and one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

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“…During last years a number of DWPs for SE with constant mass suitable for problems of chemistry (infinite at the boundaries of the interval for the spatial variable) was suggested. For them analytic solutions were obtained via the confluent Heun's function (implemented in Maple) [24], [25], [26], [27], [28], [29], [30], [31], [32], [33] and the spheroidal function (implemented in Mathematica) [25], [23]. Unfortunately they are not suitable for the problem of phosphine because in this case we deal with a position dependent mass [34], [35] (see below).…”

Section: Introductionmentioning

confidence: 99%

Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule

Sitnitsky

2021

Preprint

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The reduced mass of the effective quantum particle for the inversion mode ν 2 of phosphine molecule PH 3 is known to be a position dependent one. In the present article the inversion spectrum of PH 3 is considered with the help of the Schrödinger equation (SE) with position dependent mass and corresponding modified doublewell potential. The SE is shown to be exactly solvable. The results are used for the analysis of the pertinent experimental data available in literature. We are based on the reliable value ν3895.9 cm −1 ) obtained by Špirko et al. Also we use the value for the barrier height E b = 12300 cm −1 that seems to be commonly accepted at present and the hypothetical value for the energy splitting of the 11-th doublet of the 10ν 2 band s 10 = E 21 − E 20 ≈ 7.2 cm −1 suggested in the literature. Definite predictions are derived for the energy splitting of the 4-th doublet of the 3ν 2 band s 3 = E 7 − E 6 that is a test one for the observation in the experiment of Okuda et al. SE with position dependent mass provides selfconsistently the required values of {ν 2 ; E b ; s 10 } yielding s 3 = 6.21 • 10 −12 cm −1 .

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Exact solutions of a quartic potential

Cited by 31 publications

References 44 publications

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

Quantum Rajeev-Ranken model as an anharmonic oscillator

Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule

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