q-deformed counterpart of perturbed (λ-) hypergeometric opers

Identify and construct q-difference analogs of the perturbed (λ-) hypergeometric opers introduced in Bazhanov–Lukyanov, and characterize their properties and role in the q-deformed ODE/IM-type correspondence.

Background

The authors are motivated by the role of perturbed (λ-) hypergeometric opers in the ODE/IM correspondence in the differential setting as studied by Bazhanov and Lukyanov. Extending this structure to the q-deformed case would bridge spectral problems in quantum toroidal algebras with q-oper theory.

They explicitly point out that finding the q-deformed counterpart is an important unsolved problem reserved for future work, highlighting a concrete gap in the current oper-theoretic landscape for q-difference equations.