Nodal theorems for the Dirac equation in d >= 1 dimensions

Abstract

A single particle obeys the Dirac equation in $d \ge 1$ spatial dimensions and is bound by an attractive central monotone potential that vanishes at infinity. In one dimension, the potential is even, and monotone for $x\ge 0.$ The asymptotic behavior of the wave functions near the origin and at infinity are discussed. Nodal theorems are proven for the cases $d=1$ and $d > 1$, which specify the relationship between the numbers of nodes $n_1$ and $n_2$ in the upper and lower components of the Dirac spinor. For $d=1$, $n_2 = n_1 + 1,$ whereas for $d >1,$ $n_2 = n_1 +1$ if $k_d > 0,$ and $n_2 = n_1$ if $k_d < 0,$ where $k_d = \tau(j + \frac{d-2}{2}),$ and $\tau = \pm 1.$ This work generalizes the classic results of Rose and Newton in 1951 for the case $d=3.$ Specific examples are presented with graphs, including Dirac spinor orbits $(\psi_1(r), \psi_2(r)), r \ge 0.$