Refining the general comparison theorem for Klein-Gordon equation

Richard L. Hall; Hassan Harb

paper

Refining the general comparison theorem for Klein-Gordon equation

arXiv:2012.13008

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 $f_{1} \leq f_{2} ⟹ G_{1} (E) \leq G_{2} (E)$ , 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 $\int_{0}^{x} [f_{2} (t) - f_{1} (t)] φ_{i} (t) d t \geq 0$ , the couplings remain ordered $v_{1} \leq v_{2}$ where $i = 1 or 2,$ 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.

16 pages and 9 figures. arXiv admin note: text overlap with arXiv:1906.08762