d维Dirac方程的精化比较定理
Refined comparison theorems for the Dirac equation in d dimensions
摘要：
单个自旋$\frac{1}{2}$粒子在$d\ge 1$空间维度上服从狄拉克方程，并被一个吸引的中心单调势束缚，该势在无穷远处消失（在一维中势是偶数）。本文改进了Hall \cite{p75}的相对论比较定理。新的定理允许两个比较势$V_a$和$V_b$的图形以受控的方式交叉，并且仍然暗示每个角动量子空间底部的本征值的谱排序$E_a\le E_b$。更具体地说，在最简单的情况下，我们有：在$d=1$维中，如果$\int_0^x (V_b(t)-V_a(t)) dt\ge 0,\ x\in [0,\ \infty)$，则$E_a\le E_b$;在$d>1$维中，如果$\int_0^r (V_b(t)-V_a(t))t^{2|k_d|} dt\ge 0,\ r\in [0,\ \infty)$，其中$k_d=τ\left(j+\frac{d-2}{2}\right)$和$τ=\pm 1$，则$E_a\le E_b$。
英文摘要：
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 $k_d=τ\left(j+\frac{d-2}{2}\right)$ and $τ=\pm 1$, then $E_a\le E_b$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
