  We generalize the conjectured connection between quantum spectral problems
and topological strings to many local almost del Pezzo surfaces with arbitrary
mass parameters. The conjecture uses perturbative information of the
topological string in the unrefined and the Nekrasov-Shatashvili limit to solve
non-perturbatively the quantum spectral problem. We consider the quantum
spectral curves for the local almost del Pezzo surfaces of $\mathbb{F}_2$,
$\mathbb{F}_1$, the blowup of $\mathbb{P}^2$ in two points and a mass
deformation of the $E_8$ del Pezzo corresponding to different deformations of
the three-term operators $\mathsf{O}_{1,1}$, $\mathsf{O}_{1,2}$ and
$\mathsf{O}_{2,3}$. To check the conjecture, we compare the predictions for the
spectrum of these operators with numerical results for the eigenvalues. We also
compute the first few fermionic spectral traces from the conjectural spectral
determinant, and we compare them to analytic and numerical results in spectral
theory. In all these comparisons, we find that the conjecture is fully
validated with high numerical precision. For local $\mathbb{F}_2$ we expand the
spectral determinant around the orbifold point and find intriguing relations
for Jacobi theta functions. We also give an explicit map between the geometries
of $\mathbb{F}_0$ and $\mathbb{F}_2$ as well as a systematic way to derive the
operators $\mathsf{O}_{m,n}$ from toric geometries.
