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The topological string/spectral theory correspondence establishes a precise, non-perturbative duality between topological strings on local Calabi-Yau threefolds and the spectral theory of quantized mirror curves. This duality has been rigorously formulated for the closed string sector, but the open string sector is less understood. In this talk, I will explain how one can use open string partition functions to construct true eigenfunctions for the quantized mirror curve of local F0. We will then discuss the four-dimensional limit, underlining the implications of the topological string/spectral theory correspondence for spectral problems relating four-dimensional supersymmetric gauge theories to the quantization of their Seiberg-Witten curves
