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Nodal theorems for the Dirac equation in d ≥ 1 dimensions

A single particle obeys the Dirac equation in d≥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≥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 n1 and n2 in the upper and lower components of the Dirac spinor. For d=1 , n2=n1+1, whereas for d>1, n2=n1+1 if kd>0, and n2=n1 if kd<0, where kd=τ(j+d−22), and τ=±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 (ψ1(r),ψ2(r)),r≥0.