  We study a class of quantum integrable systems derived from dimer graphs and
also described by local toric Calabi-Yau geometries with higher genus mirror
curves, generalizing some previous works on genus one mirror curves. We compute
the spectra of the quantum systems both by standard perturbation method and by
Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way,
we obtain some exact analytic results for the classical and quantum periods of
the Calabi-Yau geometries. We also determine the differential operators of the
quantum periods and compute the topological string free energy in
Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other
methods such as the topological vertex.
