Q-functions for lambda opers

We consider the Schrödinger operators which are constructed from the λ-opers corresponding to solutions of the sl^2 Gaudin Bethe Ansatz equations. We define and study the connection coefficients called the Q-functions. We conjecture that the Q-functions obtained from the λ-opers coincide with the Q-functions of the Bazhanov–Lukyanov–Zamolodchikov opers with the monster potential related to the quantum KdV flows. We give supporting evidence for this conjecture. In particular, we give a rigorous proof that the Q-functions of λ-opers satisfy the QQ and TQ relations.