This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below.

This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in quantum field theory) should be thought of as part of the so-called trans-series expansion, containing both perturbative and non-perturbative corrections: $$\langle \mathcal{O} \rangle = \sum_{k=0}^{\infty} c_{0,k} g^k+ \sum_{I} e^{-\frac{S_I}{g^2}} \left( \sum_{k=0}^{\infty} c_{I,k} g^k \right)+ \dots \;, $$ where $I$ is a label for the saddle points of the action $S$, and $S_I$ denotes the value of the action at the $I$-th saddle point; the coefficients $c_{I,k}$ represent the perturbative expansions around the $I$-th saddle point. Assuming we know all the saddle points contributing to the path integral, all the coefficients $c_{0,k}$ and $c_{I,k}$ can be computed by perturbative methods, and we obtain a trans-series. Now we hope to turn the trans-series into a well-defined function $\langle \mathcal{O} \rangle(g)$ as a function of the coupling constant with the help of the resurgence theory. If that function is resurgent, we may adiabatically continue back to the large value of the coupling constant along a suitable choice of path in the complex plane.

Questions:(1). What are the precise mathematical conditions/criteria to turn the trans-series above into a

well-definedfunction as a function of the coupling constant with the help of the resurgence theory?When we say that the function is resurgent, does it require infinite differentiable or an infinite continuable?

p.s. The resurgence in the physics literature may refer to a stronger statement. A large-order asymptotic growth (as $k$ large) of the perturbative coefficients $c_{0,k}$ around the trivial saddle point contains the information of the non-perturbative saddle points within the same topological sector.

(2). What well-controlled mathematical properties can we state precisely for the above function $\langle \mathcal{O} \rangle$? What are the mathematical rigorous results for the re-summation and resurgence?

The context is partially inspired by this post.

J. E ́calle, Les fonctions r ́esurgentes. Tome I, vol. 5 of Publications Math ́ematiques d’Orsay 81 [Mathematical Publications of Orsay 81]. Universit ́e de Paris-Sud, D ́epartement de Math ́ematique, Orsay, 1981. Les alg`ebres de fonctions r ́esurgentes. [The algebras of resurgent functions], With an English foreword.

J. E ́calle, Les fonctions r ́esurgentes. Tome II, vol. 6 of Publications Math ́ematiques d’Orsay 81 [Mathematical Publications of Orsay 81]. Universit ́e de Paris-Sud, D ́epartement de Math ́ematique, Orsay, 1981. Les fonctions r ́esurgentes appliqu ́ees `a l’it ́eration. [Resurgent functions applied to iteration].

O. Costin, Asymptotics and Borel summability, vol. 141 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. CRC Press, Boca Raton, FL, 2009.

C. Mitschi and D. Sauzin, Divergent series, summability and resurgence. I, vol. 2153 of Lecture Notes in Mathematics. Springer, [Cham], 2016. Monodromy and resurgence, With a foreword by Jean-Pierre Ramis and a preface by E ́ric Delabaere, Mich`ele Loday-Richaud, Claude Mitschi and David Sauzin.