Proper incorporation of self-adjoint extension method to Green's function formalism : one-dimensional $\delta^{'}$-function potential case

TL;DR

This paper develops a comprehensive Green's function approach for one-dimensional -function potentials, emphasizing the role of self-adjoint extensions in defining boundary conditions and linking them to physical parameters.

Contribution

It introduces a general Green's function formulation incorporating four real self-adjoint extension parameters for the -function potential, without using perturbation theory.

Findings

Constructed the most general Green's function with four extension parameters.

Derived the relation between bare coupling constant and extension parameters.

Highlighted the importance of boundary conditions in the Green's function formalism.

Abstract

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