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We generalize the conjectured connection between quantum spectral problemsand topological strings to many local almost del Pezzo surfaces with arbitrarymass parameters. The conjecture uses perturbative information of thetopological string in the unrefined and the Nekrasov-Shatashvili limit to solvenon-perturbatively the quantum spectral problem. We consider the quantumspectral 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 massdeformation of the $E_8$ del Pezzo corresponding to different deformations ofthe 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 thespectrum of these operators with numerical results for the eigenvalues. We alsocompute the first few fermionic spectral traces from the conjectural spectraldeterminant, and we compare them to analytic and numerical results in spectraltheory. In all these comparisons, we find that the conjecture is fullyvalidated with high numerical precision. For local $\mathbb{F}_2$ we expand thespectral determinant around the orbifold point and find intriguing relationsfor Jacobi theta functions. We also give an explicit map between the geometriesof $\mathbb{F}_0$ and $\mathbb{F}_2$ as well as a systematic way to derive theoperators $\mathsf{O}_{m,n}$ from toric geometries.

Exact solutions to quantum spectral curves by topological string theory