Quantization scheme for modular $q$-difference equations

We consider modular pairs of certain second-order $q$-difference equations. An example of such a pair is the $t$-$Q$ Baxter equations for the quantum relativistic Toda lattice in the strong coupling regime. Another example from quantum mechanics is $q$-deformation of the Schrödinger equation with a hyperbolic potential. We show that the analyticity condition for the wave function or the Baxter function leads to a set of transcendental equations for the coefficients of the potential or the transfer matrix, the solution of which is their discrete spectrum.