Second the Gell-Mann and Low theorem, if a quantum system has a hamiltnian $H = H_{0} + V$, and $H_{0} | \Phi_{0} \rangle = E_{0} | \Phi_{0} \rangle$, then the following quantity

$$ | \Psi \rangle = \lim_{\epsilon \rightarrow 0} \frac{U_{\epsilon}(0,-\infty) | \Psi_{0} \rangle}{\langle \Phi_{0} | U_{\epsilon}(0, -\infty) | \Phi_{0} \rangle } $$

is a eigenstate of $H$. Recently, I have read that this result is true only if the state $\Phi_{0}$ is nondegenerated, but following the proofs in some books, like "Quantum Theory of Many-Particle Systems" of Fetter and Walecka, this hypoteshis doesn't look like necessary. Can someone explain me why?

  • $\begingroup$ Have recently read where? Which page? $\endgroup$
    – Qmechanic
    Nov 28, 2023 at 23:29


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