Non-perturbative spectra, quantum curves and mirror symmetry

"Quantum curves'' have been all the rage for various subsectors of the geometry/mathematical physics community for the last few years; yet they might mean different things for different people. I will focus on one and only one angle of this story, due to Grassi--Hatsuda--Kashaev--Marino--Zakany: in their setting, "quantum curves" is the monicker of a precise connection between the spectral theory of a class of difference operators on the real line with trace-class resolvent, and the enumerative geometry (GW/DT invariants) of toric threefolds with trivial canonical bundle. That's both bizarre -- the two subjects have a priori little overlap, and share few-to-none joint practitioners -- and deep/beautiful/powerful to various degrees. I will first review part of the existing depth/beauty/power of this story, and then outline some work in progress aimed at applying mirror-symmetry techniques (in particular, the theory of mutations of potentials of Galkin--Usnich) to the computation of quantum mechanical spectra.