论文标题
完善klein-gordon方程的一般比较定理
Refining the general comparison theorem for Klein-Gordon equation
论文作者
论文摘要
By recasting the Klein--Gordon equation as an eigen-equation in the coupling parameter $v > 0,$ the basic Klein--Gordon comparison theorem may be written $f_1\leq f_2\implies G_1(E)\leq G_2(E)$, where $f_1$ and $f_2$, are the monotone non-decreasing shapes of two central potentials $ v_1（r）= v_1 \，f_1（r）$和$ v_2（r）= v_2 \，f_2（r）$ on $ [0，\ infty）$。同时，$ v_1 = g_1（e）$和$ v_2 = g_2（e）$是相应的耦合参数，它是能量$ e \ in（-m，\，m）$的函数。我们通过证明$ \ int_0^x \ big [f_2（t） - f_1（t） - f_1（t）\ big]φ_i（t）φ_i（t）dt \ geq 0 $，削弱了地面频谱顺序的足够条件（例如$ d = 1 $尺寸）。 2，$和$ \ {φ_1，φ_2\} $分别是与耦合$ \ {v_1，\，\，v_2 \} $的地面，对于给定的$ e \ in（-m，\，m）。$。该结果扩展到$ d> 1 $尺寸的球面对称径向电位。
By recasting the Klein--Gordon equation as an eigen-equation in the coupling parameter $v > 0,$ the basic Klein--Gordon comparison theorem may be written $f_1\leq f_2\implies G_1(E)\leq G_2(E)$, where $f_1$ and $f_2$, are the monotone non-decreasing shapes of two central potentials $V_1(r) = v_1\,f_1(r)$ and $V_2(r) = v_2\, f_2(r)$ on $[0,\infty)$. Meanwhile $v_1 = G_1(E)$ and $v_2 = G_2(E)$ are the corresponding coupling parameters that are functions of the energy $E\in(-m,\,m)$. We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in $d=1$ dimension) that if $\int_0^x\big[f_2(t) - f_1(t)\big]φ_i(t)dt\geq 0$, the couplings remain ordered $v_1 \leq v_2$ where $i = 1\, {\rm or}\, 2, $ and $\{φ_1, φ_2\}$ are the ground-states corresponding respectively to the couplings $\{v_1,\, v_2\}$ for a given $E \in (-m,\, m).$. This result is extended to spherically symmetric radial potentials in $ d > 1 $ dimensions.
