Topological strings and quantum spectral problems

We consider certain quantum spectral problems appearing in the study of local Calabi-Yau geometries. The quantum spectrum can be computed by the Bohr-Sommerfeld quantization condition for a period integral. For the case of small Planck constant, the periods are computed perturbatively by deformation of the Ω background parameters in the Nekrasov-Shatashvili limit. We compare the calculations with the results from the standard perturbation theory for the quantum Hamiltonian. There have been proposals in the literature for the non-perturbative contributions based on singularity cancellation with the perturbative contributions. We compute the quantum spectrum numerically with some high precisions for many cases of Planck constant. We find that there are also some higher order non-singular non-perturbative contributions, which are not captured by the singularity cancellation mechanism. We fix the first few orders formulas of such corrections for some well known local Calabi-Yau models.