Consider the Laplace-Beltrami operator $\Delta_g$ on compact Riemannian manifold $(M,g)$, then $e^{it\sqrt{-\Delta_g}}f$ is the solution of the following Cauchy problem. $$ i\partial_tu=\sqrt{-\Delta_g}u, ~u|_{t=0}=f. $$ When $t$ is small, we have $$ e^{it\sqrt{-\Delta_g}}=Q(t)+R(t) $$ where the kernel of $Q(t)$ can be expressed by an oscillatory integral, and the reminder term has smooth kernel. My question is about what happens for large $t$, I'm not expecting good results for general manifold, however, if we just assume that $M$ is the sphere $S^{n}$ with the standard metric, can we say more about $e^{it\sqrt{-\Delta_g}}$ for large $t$? Especially, I want to know if there are explicit expressions of its kernel at any time $t>0$. Since the geodesic flow on the sphere is periodic with a common minimal period $2\pi$, is there also some periodic property of the group $e^{it\sqrt{-\Delta_g}}$？
Thanks very much for any comment or reference.
