Proper incorporation of the self-adjoint extension method to the Green function formalism: one-dimensional -function potential case

Abstract

The one-dimensional -function potential is discussed in the framework of the Green function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green function for this case is crucially dependent on the boundary conditions which are provided by the self-adjoint extension method. The most general Green function which contains four real self-adjoint extension parameters is constructed. The relation between the bare coupling constant and the self-adjoint extension parameter is also derived.