Refined comparison theorems for the Dirac equation in d dimensions

Abstract

A single spin-$\frac{1}{2}$ particle obeys the Dirac equation in $d\ge 1$ spatial dimension and is bound by an attractive central monotone potential which vanishes at infinity (in one dimension the potential is even). This work refines the relativistic comparison theorems which were derived by Hall \cite{p75}. The new theorems allow the graphs of the two comparison potentials $V_a$ and $V_b$ to crossover in a controlled way and still imply the spectral ordering $E_a\le E_b$ for the eigenvalues at the bottom of each angular momentum subspace. More specifically in a simplest case we have: in dimension $d=1$, if ${\int }_0^x(V_b(t)-V_a(t))dt\ge 0,x\in [0,\infty )$, then $E_a\le E_b$; and in $d>1$ dimensions, if ${\int }_0^r(V_b(t)-V_a(t))t^{2|k_d|}dt\ge 0,r\in [0,\infty )$, where $k_d=\tau \left(j+\frac{d-2}{2}\right)$ and $\tau =\pm 1$, then $E_a\le E_b$.

19 pages, 10 figures

References in corpus (6)

• A general comparison theorem
• Special comparison theorem for the Dirac equation
• Comparison theorems for the Klein-Gordon equation in d dimensions
• su(1,1) Algebraic approach of the Dirac equation with Coulomb-type scalar and vector potentials in D + 1 dimensions
• Comparison theorems for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions
• Nodal theorems for the Dirac equation in d >= 1 dimensions

Cited by in corpus (1)

• General comparison theorems for the Klein-Gordon equation in d dimensions