Title: WKB solutions of difference equations and reconstruction by the topological recursion,
Abstract: The purpose of this article is to analyze the connection between
Eynard-Orantin topological recursion and formal WKB solutions of a
$\hbar$-difference equation: $\Psi(x+\hbar)=\left(e^{\hbar\frac{d}{dx}}\right)
\Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$.
In particular, we extend the notion of determinantal formulas and topological
type property proposed for formal WKB solutions of $\hbar$-differential systems
to this setting. We apply our results to a specific $\hbar$-difference system
associated to the quantum curve of the Gromov-Witten invariants of
$\mathbb{P}^1$ for which we are able to prove that the correlation functions
are reconstructed from the Eynard-Orantin differentials computed from the
topological recursion applied to the spectral curve $y=\cosh^{-1}\frac{x}{2}$.
Finally, identifying the large $x$ expansion of the correlation functions,
proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new
generating series for Gromov-Witten invariants of $\mathbb{P}^1$.