We propose a construction for the eigenfunctions of a class of finite difference equations. These difference equations arise as quantizations of the mirror curves of toric Calabi-Yau threefolds, and the eigenfunctions are given in terms of refined topological string partition functions. A distinguishing feature is that these eigenfunctions are entire functions solving the difference equation for any value of the parameters, and they become eigenfunctions in the Hilbert space sense for a discrete set of eigenvalues. As such, this is some partial progress in the direction of the open topological string/spectral theory correspondence. We also investigate two interesting limits of our proposal, which give a relation between four-dimensional, N = 2, supersymmetric Yang-Mills theory (SYM) and a corresponding spectral problem. One limit gives the eigenfunctions of the quantized Seiberg-Witten curve of SU(N) SYM in terms of defect partition functions in the Nekrasov-Shatashvili limit. The other limit gives the eigenfunctions of a particular integral kernel in terms of SU(2) SYM in the self-dual phase.