General comparison theorems for the Klein-Gordon equation in d dimensions
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, H3G 1M8, Montréal, Québec, Canada
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Accepted: 19 June 2019
Published online: 17 September 2019
We study bound-state solutions of the Klein-Gordon equation , for bounded vector potentials which in one spatial dimension have the form , where is the shape of a finite symmetric central potential that is monotone non-decreasing on and vanishes as . Two principal results are reported. First, it is shown that the eigenvalue problem in the coupling parameter v leads to spectral functions of the form which are concave, and at most uni-modal with a maximum near the lower limit of the eigenenergy . This formulation of the spectral problem immediately extends to central potentials in spatial dimensions. Secondly, for each of the dimension cases, and , a comparison theorem is proven, to the effect that if two potential shapes are ordered , then so are the corresponding pairs of spectral functions for each of the existing eigenvalues. These results remove the restriction to positive eigenvalues necessitated by earlier comparison theorems for the Klein-Gordon equation.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature, 2019