Gell-Mann and Low formula for degenerate unperturbed states

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 $H_0$.

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