Let be an eigenstate of with energy and let the 'interacting' Hamiltonian be , where is a coupling constant and the interaction term. We define a Hamiltonian which effectively interpolates between and in the limit and . Let denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as of
exists, then are eigenstates of .
Note that when applied to, say, the ground-state, the theorem does not guarantee that the evolved state will be a ground state. In other words, level crossing is not excluded.
As in the original paper, the theorem is typically proved making use of Dyson's expansion of the evolution operator. Its validity however extends beyond the scope of perturbation theory as has been demonstrated by Molinari. We follow Molinari's method here. Focus on and let . From Schrödinger's equation for the time-evolution operator
and the boundary condition we can formally write
Focus for the moment on the case . Through a change of variables we can write
We therefore have that
This result can be combined with the Schrödinger equation and its adjoint
to obtain
The corresponding equation between is the same. It can be obtained by pre-multiplying both sides with , post-multiplying with and making use of
The other case we are interested in, namely can be treated in an analogous fashion
and yields an additional minus sign in front of the commutator (we are not concerned here with the case where
have mixed signs). In summary, we obtain
We proceed for the negative-times case. Abbreviating the various operators for clarity
Now using the definition of we differentiate and eliminate derivatives using the above expression, finding
where . We can now let as by assumption the in left hand side is finite. We then clearly see that is an eigenstate of and the proof is complete.