Gell-Mann and Low formula for degenerate unperturbed states

TL;DR

This paper extends the Gell-Mann and Low formula to handle degenerate unperturbed states, allowing the transformation of degenerate eigenstates into eigenstates of the perturbed Hamiltonian under certain conditions.

Contribution

It introduces a method to apply the Gell-Mann and Low switching to degenerate states by focusing on eigenstates of a finite rank operator within the degenerate subspace.

Findings

Extension of Gell-Mann and Low formula to degenerate states

Conditions under which the switching procedure remains valid

Application to eigenstates of a finite rank operator

Abstract

The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian $H_0$ into eigenstates of the modified Hamiltonian $H_0 + V$. This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator $\cP_0 V \cP_0$, where $\cP_0$ is the projection onto a degenerate eigenspace of $H_0$.