Refining the general comparison theorem for the Klein–Gordon equation

Abstract

By recasting the Klein–Gordon equation as an eigen-equation in the coupling parameter [Formula: see text] the basic Klein–Gordon comparison theorem may be written [Formula: see text], where [Formula: see text] and [Formula: see text] are the monotone nondecreasing shapes of two central potentials [Formula: see text] and [Formula: see text] on [Formula: see text]. Meanwhile, [Formula: see text] and [Formula: see text] are the corresponding coupling parameters that are functions of the energy [Formula: see text]. We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in [Formula: see text] dimension) that if [Formula: see text], the couplings remain ordered [Formula: see text], where [Formula: see text] and [Formula: see text] are the ground-states corresponding, respectively to the couplings [Formula: see text] for a given [Formula: see text]. This result is extended to spherically symmetric radial potentials in [Formula: see text] dimensions.