Related papers: Refining the general comparison theorem for Klein-Gordon equation

Refining the general comparison theorem for Klein-Gordon equation

URL: http://arxiv.org/abs/2012.13008v1
Date: Wed, 23 Dec 2020 22:36:48 GMT
Title: Refining the general comparison theorem for Klein-Gordon equation
Authors: Richard L. Hall, Hassan Harb
Abstract summary: We recast the Klein-Gordon-Gordon equation as an eigen-equation in the coupling parameter $v > 0,$ the basic Klein-Gordon comparison theorem. We weaken the sufficient condition for the groundstate spectral ordering by proving. that if $intxbig[f_2(t)big]varphi_i(t)dtgeq 0$, the couplings remain ordered.
Score: 2.4366811507669124
License: http://creativecommons.org/licenses/by/4.0/
Abstract: 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 $\int_0^x\big[f_2(t) - f_1(t)\big]\varphi_i(t)dt\geq 0$, the couplings remain ordered $v_1 \leq v_2$ where $i = 1\, {\rm or}\, 2, $ and $\{\varphi_1, \varphi_2\}$ 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.

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