Refining the general comparison theorem for Klein-Gordon equation
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 f1≤ f2 G1(E)≤ G2(E), where f1 and f2, are the monotone non-decreasing shapes of two central potentials V1(r) = v1\,f1(r) and V2(r) = v2\, f2(r) on [0,∞). Meanwhile v1 = G1(E) and v2 = G2(E) are the corresponding coupling parameters that are functions of the energy E∈(-m,\,m). We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in d=1 dimension) that if ∫0x[f2(t) - f1(t)]i(t)dt≥ 0, the couplings remain ordered v1 ≤ v2 where i = 1\, or\, 2, and \1, 2\ are the ground-states corresponding respectively to the couplings \v1,\, v2\ for a given E ∈ (-m,\, m).. This result is extended to spherically symmetric radial potentials in d > 1 dimensions.