  We consider the Schr\"odinger 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.
