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arXiv · 0811.1222

Comparison theorems for the Klein-Gordon equation in d dimensions

Richard L. Hall·M. D. S. Aliyu
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Abstract

Two comparison theorems are established for discrete eigenvalues of the Klein-Gordon equation with an attractive central vector potential in d >= 1 dimensions. (I) If ψ_1 and ψ_2 are node-free ground states corresponding to positive energies E_1 >= 0 and E_2 >= 0, and V_1(r) <= V_2(r) <= 0, then it follows that E_1 <= E_2. (II) If V(r,a) depends on a parameter a \in(a_1,a_2), V(r,a) <= 0, and E(a) is any positive eigenvalue, then \partial V/\partial a >= 0 ==> E'(a) >= 0 and \partial V/\partial a <= 0 ==> E'(a) <= 0.

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Richard L. Hall, M. D. S. Aliyu. 2008-11-07. Comparison theorems for the Klein-Gordon equation in d dimensions. https://doi.org/10.1103/physreva.78.052115

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