Exact eigenfunctions and the open topological string

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.