Absence of Cooper-type bound states in three- and few-electron systems

TL;DR

This paper demonstrates that unlike the two-electron Cooper problem, three- and few-electron systems on the full Fermi sea do not form Cooper-type bound states due to the absence of a key singularity in their kernel.

Contribution

It reveals the absence of Cooper-type bound states in three- and few-electron systems, contrasting with the two-electron case, and clarifies the role of the kernel singularity.

Findings

No Cooper-type bound states in three- and few-electron systems

Absence of the fixed-point singularity in the kernel

Difference from the two-electron Cooper problem

Abstract

It is shown that the appearance of a fixed-point singularity in the kernel of the two-electron Cooper problem is responsible for the formation of the Cooper pair for an arbitrarily weak attractive interaction between two electrons. This singularity is absent in the problem of three and few superconducting electrons at zero temperature on the full Fermi sea. Consequently, such three- and few-electron systems on the full Fermi sea do not form Cooper-type bound states for an arbitrarily weak attractive pair interaction.