Comparison theorems for the Klein-Gordon equation in d dimensions

TL;DR

This paper establishes comparison theorems for eigenvalues of the Klein-Gordon equation with attractive potentials in multiple dimensions, providing conditions under which eigenvalues increase or decrease based on potential variations.

Contribution

It introduces two new comparison theorems for discrete eigenvalues of the Klein-Gordon equation with central potentials in arbitrary dimensions.

Findings

Eigenvalues increase with more negative potentials for ground states.

Eigenvalues vary monotonically with potential parameters under certain conditions.

Theorems apply to node-free ground states with positive energies.

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 \psi_1 and \psi_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.