Green Function Theory and its Global Structure

Abstract

The Green function theory is successful in many fields of theoretical physics and the Bogoliubov-Gor'kov theory system is its important branch that involves many theoretical methods. In the last twenty years many papers have been published, but as all the equations of motion are coupled and their number is infinite, its global structure problem has not been investigated as yet. This paper is devoted to the study of its global structure, the exact decoupling problem, and the uniqueness and completeness problems. Some higher order spectral representation theorems and exact relationships between higher and lower order Green functions are obtained. Thus the equations of motion are decoupled exactly.In this paper it is proved that after cutting off the equations of motion are decoupled exactly and the solution of these equation systems may not be unique. It is also proved that if there is one solution satisfying all the Green function's equations, then there must be a solution set, all of them satisfy all the equations and its number is infinite and the differences between them are arbitrary.By adding some limitations to ImG(x) at the real axis a uniqueness theorem is proved.