Green’s functions for nonlinear Klein–Gordon kink perturbation theory

A Green’s function is defined for nonlinear Klein–Gordon theories in terms of the solutions to the eigenvalue equation obtained by linearizing the nonlinear wave equation about a static kink waveform. Analytic forms in terms of ‘‘modified’’ Lommel functions of two variables are derived for the sine–Gordon, phi-4, and double quadratic potentials. Asymptotic forms for the Green’s functions are obtained by investigating the asymptotic behavior of the modified Lommel functions. Methods for calculating the Lommel functions are also outlined.