This question originates from reading the [proof of Gell-Mann Low thoerem](https://arxiv.org/abs/math-ph/0612030v1).
Let |ψ0⟩ be an eigenstate of H0 with eigenvalue E0, and consider the state vector
|ψ(−)ϵ⟩=Uϵ,I(0,−∞)|ψ0⟩⟨ψ0|Uϵ,I(0,−∞)|ψ0⟩
**Gell-Mann and Low's theorem:**
If the |ψ(−)⟩:=lim exist, then |\psi^{(-)} \rangle must be an eigenstate of H with eigenvalue E. And the eigenvalue E is decided by following equation:
\Delta E= E-E_0=-\lim_{\epsilon\rightarrow 0^+} i\epsilon g\frac{\partial}{\partial g}\ln \langle\psi_0| U_{\epsilon,I}(0,-\infty)|\psi_0\rangle
However we learn in scattering theory,
U_I(0,-\infty) = \lim_{t\rightarrow -\infty} U_{full}(0,t)U_0(t,0) = \Omega_{+}
where \Omega_{+} is the Møller operator. We can prove the identity H\Omega_{+}=H_0 \Omega_{+} for the Møller operator. It says that the energy of a scattering state will not change when you turn on the interaction adiabatically.
My questions:
1.The only way to avoid these contradiction is to prove that \Delta E for scattering state of H_0 must be zero. How to prove this? In general, it should be that for a scattering state there will be no energy shift, for discrete state there will be some energy shift. But the Gell-Mann Low theorem do not tell me the result.es
2.How to tackle this explicit case?
H_0 = \frac{\mathbf{p}^2}{2}- \frac{1}{|\mathbf{r}|}, \ H_I= \frac{1}{|\mathbf{r}|}
then H=H_0+H_I=\frac{\mathbf{p}^2}{2}. If we start with |\psi_0\rangle is the ground state of H_0, i.e. the ground state of Hydrogen atom, and evolve by U_{ I}(0,-\infty)|\psi_0\rangle , what is the result of this state? Although in this case the system experience some level crossing, the theorem tells us if the state exists, then it must be some eigenstate of H. In this case the adiabatic theorem cannot be used, but the Gell-mann Low theorem still works.
3.The existence of \lim_{\epsilon\rightarrow 0^{+}}|\psi^{(-)}_\epsilon\rangle is annoying. Is there some criterion of existence of \lim_{\epsilon\rightarrow 0^{+}}|\psi^{(-)}_\epsilon\rangle? Or give me an explicit example in which this does exixt.
4.It seems that the Gell-Mann Low theorem is a generalized adiabatic theorem, which can be used for discrete spectrum or continuous spectrum. How to prove Gell-Mann Low theorem reduces to the adiabatic theorem under the condition of the adiabatic theorem?