  This work develops a new method to calculate non-perturbative corrections in
one-dimensional Quantum Mechanics, based on trans-series solutions to the
refined holomorphic anomaly equations of topological string theory. The method
can be applied to traditional spectral problems governed by the Schr\"odinger
equation, where it both reproduces and extends the results of well-established
approaches, such as the exact WKB method. It can be also applied to spectral
problems based on the quantization of mirror curves, where it leads to new
results on the trans-series structure of the spectrum. Various examples are
discussed, including the modified Mathieu equation, the double-well potential,
and the quantum mirror curves of local $\mathbb{P}^2$ and local $\mathbb{F}_0$.
In all these examples, it is verified in detail that the trans-series obtained
with this new method correctly predict the large-order behavior of the
corresponding perturbative sectors.
