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

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Hall R. L., Yesiltas Ö.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, vol.43, no.19, 2010 (SCI-Expanded) identifier identifier


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)) <= V((2)), then corresponding discrete eigenvalues are all similarly ordered E(kappa nu)((1)) <= E(kappa nu)((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.