I'm reading chapter 4 of Introduction to Quantum Field Theory by Peskin & Schroeder. In the $\phi^4$ theory, the authors state that the ground state of the interaction theory $|\Omega\rangle$ can be written as ($\hbar=1$ )
$$|\Omega\rangle=\lim_{T\to\infty(1-i\epsilon)}\left( e^{-iE_{0}T} \langle \Omega | 0 \rangle \right)^{-1}e^{-iHT}|0\rangle$$
where $E_{0}=\langle\Omega|H|\Omega\rangle$,$|0\rangle$ is the free theory vacuum and $E_n, |n\rangle$ are the eigenvalues and eigenstates of the hamiltonian $H$ .I'm trying to understand that letting the time be "slightly ($\epsilon$ small in some sense) imaginary" is the way to get rid of $n\neq0$ terms in the following series, via a decaying real exponential:
$$e^{-iHT}|0\rangle=e^{-iE_{0}T}|\Omega\rangle\langle\Omega|0\rangle\ + \sum_{n\neq0}e^{-iE_{n}T}|n\rangle\langle n|0\rangle $$
$$ \sum_{n\neq0}e^{-iE_{n}T(1-i\epsilon)}|n\rangle\langle n|0\rangle= \sum_{n\neq0}e^{-iE_{n}T}|n\rangle\langle n|0\rangle+ \sum_{n\neq0}e^{-\epsilon E_{n} T}|n\rangle\langle n|0\rangle$$
Question 1) I don't really see the oscillating term can be neglected as $T \to \infty$
Question 2) How can be physically justified the following substitution $T\to T-i\epsilon$ ?
Thanks for your time