Green function formulation of quantum mechanics Suppose that I am given the vacuum expectation value of time ordered products of position operators in Heisenberg picture. Given this Green's function, is it possible to obtain information about the energy eigenvalues by inserting a complete set of basis states? I am attaching the screenshot of the problem below:

I tried inserting the states, but could not figure out how we could gain information about the eigenvalues (and ultimately about the functional form of the Hamiltonian) if we are only given the Green functions.
Note: This is actually problem 1 of chapter 3 in Tom Bank's book on qft.
 A: I think it should be possible to recover the energy eigenvalues just from the 2-points function. For $t_2 \geq t_1$, using that $H \left|0\right\rangle = 0$, I have:
$$
\left\langle 0 \middle| \,\hat{x}(t_2) \, \hat{x}(t_1)\, \middle| 0 \right\rangle
=
\left\langle 0 \middle| \,\hat{x} \,e^{i \delta t H}\, \hat{x}\, \middle| 0 \right\rangle
=
\sum_n e^{i \delta t E_n}\, \left\langle 0 \middle| \,\hat{x} \, \middle| n \right\rangle \,\left\langle n \middle| \hat{x}\, \middle| 0 \right\rangle
=: G(\delta t)
$$
where $\delta t = t_2 - t_1 \geq 0$, $\hat{x} := \hat{x}(0)$ and $\left| n \right\rangle, E_n$ are the energy eigenvectors/eigenvalues. Since $\alpha_n := \left\langle 0 \middle| \,\hat{x} \, \middle| n \right\rangle \,\left\langle n \middle| \hat{x}\, \middle| 0 \right\rangle$ is real, we have $G(-\delta t) = \overline{G(\delta t)}$, so we know $G$ for all $\delta t \in \mathbb{R}$. Then its Fourier transform will be a sum of Dirac deltas localized at the $E_n$'s, with amplitudes $\alpha_n$.
I guess more information can be gained by looking at $3$-points functions and so on, ultimately recovering the full information on the theory from its Green functions.
To give a bit of context, this problem can be seen as a baby version of the reconstruction of a QFT from its path integral (à la Osterwalder–Schrader reconstruction theorem).
