quantum mechanics - The contradiction between Gell-mann Low theorem and the identity of Møller operator $HOmega_{+}=Omega_{+}H_0$

This question originates from reading the proof of Gell-mann Low thoerem.

$H=H_0+H_I$, let $|\psi_0\rangle$ be an eigenstate of $H_0$ with eigenvalue $E_0$, and consider the state vector defined as

$$|\psi^{(-)}_\epsilon\rangle=\frac{U_{\epsilon,I}(0,-\infty)|\psi_0\rangle}{\langle \psi_0| U_{\epsilon,I}(0,-\infty)|\psi_0\rangle}$$ where the definition of $U_{\epsilon,I}(0,-\infty)$ can be found in the above paper

Gell-Mann and Low's theorem: If the $|\psi^{(-)} \rangle :=\lim_{\epsilon\rightarrow 0^{+}}|\psi^{(-)}_\epsilon\rangle$ 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,

$$U_I(0,-\infty) = \lim_{\epsilon\rightarrow 0^{+}} U_{\epsilon,I}(0,-\infty) = \lim_{t\rightarrow -\infty} U_{full}(0,t)U_0(t,0) = \Omega_{+}$$ where $\Omega_{+}$ is the Møller operator. We can prove the identity for Møller operator $H\Omega_{+}= \Omega_{+}H_0$ in scattering theory. It says the energy of scattering state will not change when you turn on the interaction adiabatically.

My question:

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? In general, it should be that for scattering state there will be no energy shift, for discrete state there will be some energy shift. But Gell-Mann Low theorem do not tell me the result.

2.It seems that the Gell-Mann-Low theorem is more powerful than adiabatic theorem which requires that there must exist gap around the evolving eigenstate. And Gell-Mann-Low theorem can be applied to any eigenstate of $H_0$ no matter whether the state is discrete, continous or degenerate and no matter whether there is level crossing during evolution. However the existence of $\lim_{\epsilon\rightarrow 0^{+}}|\psi^{(-)}_\epsilon\rangle$ is annoying, which heavily restrict the application of this theorem. Is there some criterion of existence of $\lim_{\epsilon\rightarrow 0^{+}}|\psi^{(-)}_\epsilon\rangle$? Or give me an explicit example in which this doesn't exixt.

3.It seems that Gell-Mann Low theorem is a generalized adiabatic theorem, which can be used in discrete spectrum or contiunous spectrum. How to prove Gell-Mann Low theorem can return to adiabatic theorem in condition of adiabatic theorem. Need to prove that the $\lim_{\epsilon\rightarrow 0^{+}}|\psi^{(-)}_\epsilon\rangle$ exist given the requirement of the adiabatic theorem.

Answer

The Gell-Mann Low theorem applies only to eigenvectors, i.e. to the discrete part of the spectrum. Hence it does not apply to scattering states. The latter are not eigenvectors since they are not normalizable. Your formula for $\Delta E$ is meaningless for them since the inner product on the right hand side is generally undefined unless $\psi_0$ is normalizable.

[The equation for the Moeller operator] ''says the energy of scattering state will not change when you turn on the interaction adiabatically.'' No. It only says that $H$ and $H_+$ must have the same total spectrum; it says nothing about energies of individual scattering states.

Moreover, a more rigorous treatment (e.g. in the math physics treatise by Thirring) shows that your equation holds at best on the subspace orthogonal to the discrete spectrum (which almost always exhibits energy shifts), and that certain assumptions (relative compact perturbations) must be satisfied that it holds on this projection. These assumptions are not satisfied when the continuous spectrum of $H$ and $H_0$ is not identical, e.g., when $H_0$ is for a free particle and $H$ for a harmonic oscillator or a Morse oscillator, or vice versa.