A very common inverse problem in mathematical physics is trying to understand the potential of a quantum mechanical system given its scattering data. Such problems, although very interesting, are very challenging and usually ill-posed. I'm trying to understand one inverse problem that I guess is particularly simple.
Some quantum mechanical models are ill-defined because their potential is unbounded from below (e.g., $\phi^3$). Still, perturbation theory is typically well-defined, as least in the sense of formal power series. As far as Feynman diagrams is concerned, the theories $\phi^3$ and $\phi^4$ are not fundamentally different, although only the second one represents an approximation to a meaningful non-perturbative theory. The same thing can be said about standard quantum mechanical models such as an anharmonic oscillator with cubic and quartic terms.
Assume we are given an arbitrarily large but finite number of terms in the perturbative expansion of, say, the partition function of a certain unknown system. (Recall that the logarithm of partition function represents the ground state energy, so it is in principle observable). Question: Can we predict whether the underlying potential of such a series is bounded from below?
In other words, a single term in the perturbative series is qualitatively identical in the bounded and unbounded case. But what if we zoom-out and look at many terms? Does the (truncated) series contain any information that allows us to tell them apart? Or is the series truly oblivious to the behaviour of the potential far from the equilibrium position?
It seems clear to me that from a truncated series we can at best predict the probability that the potential is bounded. The more terms, the better the prediction (in the sense of a confidence interval). As long as the number is finite, we can never be sure the potential is bounded or not. But is there such a probabilistic estimator at all? or is it really impossible to even predict a probability?