  In this article, we make a close analysis on quantum multiplication operators
on the quantum cohomology rings of Lagrangian and orthogonal Grassmannians, and
give an explicit description on all simultaneous eigenvectors and the
corresponding eigenvalues for these operators. As a result, we show that
Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani holds for these
manifolds.
