The complex side of the TS/ST correspondence

The TS/ST correspondence relates the spectral theory of certain quantum mechanical operators, to topological strings on toric Calabi-Yau threefolds. So far the correspondence has been formulated for real values of Planck's constant. In this paper we start to explore the validity of the correspondence when $\hbar$ takes complex values. We give evidence that, for threefolds associated to supersymmetric gauge theories, one can extend the correspondence and obtain exact quantization conditions for the operators. We also explore the correspondence for operators involving periodic potentials. In particular, we study a deformed version of the Mathieu equation, and we solve for its band structure in terms of the quantum mirror map of the underlying threefold.