  Mirror curves to toric Calabi-Yau threefolds can be quantized and lead to
trace class operators on the real line. The eigenvalues of these operators are
encoded in the BPS invariants of the underlying threefold, but much less is
known about their eigenfunctions. In this paper we first develop methods in
spectral theory to compute these eigenfunctions. We also provide a matrix
integral representation which allows to study them in a 't Hooft limit, where
they are described by standard topological open string amplitudes. Based on
these results, we propose a conjecture for the exact eigenfunctions which
involves both the WKB wavefunction and the standard topological string
wavefunction. This conjecture can be made completely explicit in the maximally
supersymmetric, or self-dual case, which we work out in detail for local P1xP1.
In this case, our conjectural eigenfunctions turn out to be closely related to
Baker-Akhiezer functions on the mirror curve, and they are in full agreement
with first-principle calculations in spectral theory.
