Refined comparison theorems for the Dirac equation with spin and pseudo-spin symmetry in d dimensions

Hall, Richard L.; Zorin, Petr

doi:10.1140/epjp/i2016-16102-y

Cited by 5 publications

(12 citation statements)

References 105 publications

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“…Since V vanishes at ∞, then equation (12) becomes ϕ = (m 2 − E 2 )ϕ near infinity, which means that |E| < m by the same reasoning as used for equation (2). Since the derivative of the term Q r 2 in equation (12) with respect to E is equal to zero, then by the same reasoning the relation (4) is also valid for all other d > 1 dimensions. We define the operator…”

Section: B D-dimensional Cases (D > 1)mentioning

confidence: 85%

General comparison theorems for the Klein-Gordon equation in d dimensions

Hall

Harb

2019

Eur. Phys. J. Plus

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We study bound-state solutions of the Klein-Gordon equationfor bounded vector potentials which in one spatial dimension have the form V (x) = v f (x), where f (x) ≤ 0 is the shape of a finite symmetric central potential that is monotone non-decreasing on [0, ∞) and vanishes as x → ∞. Two principal results are reported. First, it is shown that the eigenvalue problem in the coupling parameter v leads to spectral functions of the form v = G(E) which are concave, and at most uni-modal with a maximum near the lower limit E = −m of the eigenenergy E ∈ (−m, m). This formulation of the spectral problem immediately extends to central potentials in d > 1 spatial dimensions. Secondly, for each of the dimension cases, d = 1 and d ≥ 2, a comparison theorem is proven, to the effect that if two potential shapes are ordered f1(r) ≤ f2(r), then so are the corresponding pairs of spectral functions G1(E) ≤ G2(E) for each of the existing eigenvalues. These results remove the restriction to positive eigenvalues necessitated by earlier comparison theorems for the Klein-Gordon equation.Hall and Harb at a function v(E), whereas E(v) is a two-valued expression. In section 2, we establish the principal features of the spectral curves v(E) for the class of negative bounded potentials that vanish at infinity. In section 3 we solve the Klein-Gordon equation analytically for the square-well potential in d ≥ 1 dimensions. In section 4 we prove some comparison theorems: the principal results claim that for any discrete eigenvalue E ∈ (−m, m) and negative potential-shape functions f 1 (r) and f 2 (r) we haveIn section 5 we exhibit a complete recipe for spectral bounds for this class of potentials based on comparisons with the exactly soluble square-well problem.II. GENERAL FEATURES OF THE SPECTRAL CURVE G(E) = v(E). A. One-dimensional caseThe Klein-Gordon equation in one dimension is given by:

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Section: B D-dimensional Cases (D > 1)mentioning

confidence: 85%

General comparison theorems for the Klein-Gordon equation in d dimensions

Hall

Harb

2019

Eur. Phys. J. Plus

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“…By subtituting Eqs. (15) into Eq. (14) and using variable separation method, we get the radial part and the angular part of Dirac equations in hypersperical coordinate with D = 5.…”

Section: Dirac Equation With Separable Qdeformed Quantum Potetial In mentioning

confidence: 99%

“…The angullar part of 5-dimensional that obtained from Eqs. (14)(15) can be resolved into four parts, and for , we get…”

Section: The Angular Partmentioning

confidence: 99%

“…[3] Pseudo-spin symmetry has been observed in deformed nuclei and can be enhanced in heavy proton-rich nuclei. [4] Solution of Dirac equation for some potentials under limit case of spin symmetry and pseudo-spin symmetry have been investigated intensively whether in three- [5,6] two-or one- [7][8][9][10][11][12][13] dimensional space, and some D-dimensional with spherically Symmetric Spacetimes [14][15]. However, The D-dimensional Dirac equation with (D-1)-dimensional separable non-central potential has not been investigated yet, therefore it may be worthy to investigate Dirac equation in 5 dimensions with separable 4-dimensional noncentral potential in this study.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Relativistic Energy Analysis of Five-Dimensionalq-Deformed Radial Rosen-Morse Potential Combined withq-Deformed Trigonometric Scarf Noncentral Potential Using Asymptotic Iteration Method

Pramono

Супармі

Cari

2016

Advances in High Energy Physics

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In this work,we study the exact solution of Dirac equation in the hyper-spherical coordinate under influence of separable q-Deformed quantum potentials. The q-deformed hyperbolic Rosen-Morse potential is perturbed by qdeformed non-central trigonometric Scarf potentials, where whole of them can be solved by using Asymptotic Iteration Method (AIM). This work is limited to spin symmetry case. The relativistic energy equation and orbital quantum number equation lD-1 have been obtained using Asymptotic Iteration Method. The upper radial wave function equations and angular wave function equations are also obtained by using this method. The relativistic energy levels are numerically calculated using Mat Lab, the increase of radial quantum number n causes the increase of bound state relativistic energy level both in dimension D = 5 and D = 3. The bound state relativistic energy level decreases with increasing of both deformation parameter q and orbital quantum number nl. s s s z

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“…The idea of refining the comparison theorems of non-relativistic and relativistic quantum mechanics has been first presented in Ref. [22], and was applied for the Dirac equation [23,24], and the Klein-Gordon equation [25], but the latter case was restricted to non-negative energies. The present work removes the restriction to positive energies and the results are obtained for attractive central potentials in all dimensions d ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

Refining the general comparison theorem for Klein-Gordon equation

Hall¹,

Harb²

2020

Preprint

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By recasting the Klein-Gordon equation as an eigen-equation in the coupling parameter v > 0, the basic Klein-Gordon comparison theorem may be written f1 ≤ f2 =⇒ G1(E) ≤ G2(E), where f1 and f2, are the monotone non-decreasing shapes of two central potentials V1(r) = v1 f1(r) and V2(r) = v2 f2(r) on [0, ∞). Meanwhile v1 = G1(E) and v2 = G2(E) are the corresponding coupling parameters that are functions of the energy E ∈ (−m, m). We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in d = 1 dimension) that if x 0 f2(t) − f1(t) ϕi(t)dt ≥ 0, the couplings remain ordered v1 ≤ v2 where i = 1 or 2, and {ϕ1, ϕ2} are the ground-states corresponding respectively to the couplings {v1, v2} for a given E ∈ (−m, m).. This result is extended to spherically symmetric radial potentials in d > 1 dimensions.

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Refined comparison theorems for the Dirac equation with spin and pseudo-spin symmetry in d dimensions

Cited by 5 publications

References 105 publications

General comparison theorems for the Klein-Gordon equation in d dimensions

General comparison theorems for the Klein-Gordon equation in d dimensions

Relativistic Energy Analysis of Five-Dimensionalq-Deformed Radial Rosen-Morse Potential Combined withq-Deformed Trigonometric Scarf Noncentral Potential Using Asymptotic Iteration Method

Refining the general comparison theorem for Klein-Gordon equation

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