Solvable spectral problems from 2d CFT and $\mathcal =$ 2 gauge theories

The so-called 2d/4d correspondences connect two-dimensional conformal field theory (2d CFT), = 2 supersymmetric gauge theories and quantum integrable systems. The latter in the simplest case of the SU(2) gauge group are nothing but the quantum-mechanical systems. In the present article we summarize our recent results and list open problems concerning an application of the aforementioned dualities in the studies of spectral problems for some Schrodinger operators with Mathieu–type periodic, periodic PT–symmetric and (Heun's) elliptic potentials.