  Based on previous insights, we present an ansatz to obtain quantization
conditions and eigenfunctions for a family of difference equations which arise
from quantized mirror curves in the context of local mirror symmetry of toric
Calabi-Yau threefolds. It is a first principles construction, which yields
closed expressions for the quantization conditions and the eigenfunctions when
$\hbar/2\pi \in \mathbb Q$. The key ingredient is the modular duality structure
of the underlying quantum integrable system. We use our ansatz to write down
explicit results in some examples, which are successfully checked against
purely numerical results for both the spectrum and the eigenfunctions.
Concerning the quantization conditions, we also provide evidence that, in the
rational case, this method yields a resummation of conjectured quantization
conditions involving enumerative invariants of the underlying toric Calabi-Yau
threefold.
