Comparison theorems for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions

Abstract

A single Dirac particle is bound in d dimensions by vector V(r) and scalar S(r) central potentials. The spin-symmetric S=V and pseudo-spin-symmetric S = - V cases are studied and it is shown that if two such potentials are ordered V^{(1)} \le V^{(2)}, then corresponding discrete eigenvalues are all similarly ordered E_{κν}^{(1)} \le E_{κν}^{(2)}. This comparison theorem allows us to use envelope theory to generate spectral approximations with the aid of known exact solutions, such as those for Coulombic, harmonic-oscillator, and Kratzer potentials. The example of the log potential V(r) = v\ln(r) is presented. Since $V(r)$ is a convex transformation of the soluble Coulomb potential, this leads to a compact analytical formula for lower-bounds to the discrete spectrum. The resulting ground-state lower-bound curve E_{L}(v) is compared with an accurate graph found by direct numerical integration.

12 pages, 1 figure.

References in corpus (4)

• Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: the case of the relativistic harmonic oscillator
• Special comparison theorem for the Dirac equation
• Comparison theorems for the Klein-Gordon equation in d dimensions
• Dynamical symmetries of the Klein-Gordon equation

Cited by in corpus (1)

• Supersymmetric analysis for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions