This work explores the connection between spectral theory and topological strings. A concrete example (the Y(3,0) geometry) of a conjectured exact relation between both based on mirror symmetry (TS/ST correspondence) is analysed in detail, to find a complete agreement. This is used as motivation to apply string theory tools, and in particular the refined holomorphic anomaly, to other Schrödinger-type spectral problems. With it, we efficiently compute their all-orders WKB expansion. We also upgrade the refined holomorphic anomaly to include non-perturbative corrections to the WKB series. This is used to retrieve the transseries generated by previously known exact quantization conditions for quantum mechanical problems. Via resurgence, it will allows us to reproduce the large order behaviour of the WKB coefficients, for both the Schrödinger problems and the quantum mirror curves of the TS/ST correspondence.