  Let $M$ be a Fano manifold, and $H^\star(M;\mathbb{C})$ be the quantum
cohomology ring of $M$ with the quantum product $\star.$ For $\sigma \in
H^*(M;\mathbb{C})$, denote by $[\sigma]$ the quantum multiplication operator
$\sigma\star$ on $H^*(M;\mathbb{C})$. It was conjectured several years ago
\cite{GGI, GI} and has been proved for many Fano manifols \cite{CL1, CH2,
LiMiSh, Ke}, including our cases, that the operator $[c_1(M)]$ has a real
valued eigenvalue $\delta_0$ which is maximal among eigenvaules of $[c_1(M)]$.
Galkin's lower bound conjecture \cite{Ga} states that for a Fano manifold $M,$
$\delta_0\geq \mathrm{dim} \ M +1,$ and the equlity holds if and only if $M$ is
the projective space $\mathbb{P}^n.$ In this note, we show that Galkin's lower
bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some
exceptions for the equality.
