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

The classic comparison theorem of quantum mechanics states that if two potentials are ordered then the corresponding energy eigenvalues are similarly ordered, that is to say if $V_a\le V_b$, then $E_a\le E_b$. Such theorems have recently been established for relativistic problems even though the discrete spectra are not easily characterized variationally. In this paper we improve on the basic comparison theorem for the Dirac equation with spin and pseudo--spin symmetry in $d\ge 1$ dimensions. The graphs of two comparison potentials may now cross each other in a prescribed manner implying that the energy values are still ordered. The refined comparison theorems are valid for the ground state in one dimension and for the bottom of an angular momentum subspace in $d>1$ dimensions. For instance in a simplest case in one dimension, the condition $V_a\le V_b$ is replaced by $U_a\le U_b$, where $U_i(x)=\int_0^x V_i(t)dt$, $x\in[0,\ \infty)$, and $i=a$ or $b$.