# Gell-Mann and Low formula for degenerate unperturbed states

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 the degenerate eigenspace of $H_0$.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00394223
Contributor : Gabriel Stoltz <>
Submitted on : Thursday, June 11, 2009 - 8:38:52 AM
Last modification on : Friday, March 27, 2020 - 3:11:16 AM

### Citation

Christian Brouder, Gianluca Panati, Gabriel Stoltz. Gell-Mann and Low formula for degenerate unperturbed states. Ann. I. H. Poincare-Phy, 2010, 10 (7), pp.1285-1309. ⟨10.1007/s00023-009-0018-7⟩. ⟨hal-00394223⟩

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