We consider the Schrödinger operators which are constructed from the \(\lambda \) -opers corresponding to solutions of the \(\widehat{\mathfrak {sl}}_2\) Gaudin Bethe Ansatz equations. We define and study the connection coefficients called the Q-functions. We conjecture that the Q-functions obtained from the \(\lambda \) -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 \(\lambda \) -opers satisfy the QQ and TQ relations.
