I have been thinking lately about the following question. In quantum field theory, we define spectral density function $\rho(\mu^2)$ using two-point function as follows (Källén–Lehmann formula) $$ \langle 0|T \phi(x) \phi(y)|0\rangle = \int \limits_0^\infty d\mu^2 \rho(\mu^2) D(x-y,\mu^2) $$ where $D$ is free field propagator. The function $\rho(\mu^2)$ is said to contain information about all possible excitations in a system (single particle states are its poles, multiparticle ones correspond to branch cuts, etc,etc)
The question is as follows: is knowing the spectral density in QFT enough to specify the theory? Or, in other words, say we have two QFTs with the same spectral density function; is it possible to identify them? Say, in quantum mechanics (given boundary conditions) there is such a thing as inverse scattering problem. In a good enough situation one can reconstruct the potential if knowing the spectral measure - energy levels and their degenerateness (I am not sure the analogy here is correct, but this analogy is one of the things that make me confused about this question). Yet it seems that in QFT only knowing two-point functions are not enough; we kind of only have information about "one-particle" excitations this way. Yet, it still contains a lot of information about interactions such as bound states, energy corrections for multiparticle states and so on. Is this information enough in QFT?
I would appreciate any discussion or known result connected with this problem.