# An elliptic operator whose corresponding symbol Hamiltonian vector field has an isolated periodic orbit

Let $$D$$ be a differential operator on the space of smooth functions on a manifold $$M$$. The symbol of $$D$$ can be considered as a Hamiltonian on the cotangent bundle $$T^*M$$. We call this Hamiltonian as "Corresponding symbol Hamiltonian"

Motivated by the above interesting linked question and this post and this one we ask the following question:

Is there an elliptic operator on a manifold whose corresponding symbol Hamiltonian has an isolated periodic orbit?

Note: We add the ellipticity condition since we learn from this answer that for differential operator associated with a vector field, which is a non elliptic operator, we do not have an isolated periodic orbit

I'm going to assume that you want an isolated periodic orbit on some fixed energy level. Pick your favorite Riemannian manifold $$(M,g)$$ such that there is an isolated closed geodesic. Then, the geodesic flow on the unit tangent bundle has a corresponding isolated period orbit.
The Laplace-Beltrami operator $$\Delta_{g}$$ is elliptic and has principal symbol
$$\sigma(\Delta_{g})(\xi)=\lVert \xi \rVert_{g}^{2}.$$
as a function on $$T^{*}M.$$ The Hamiltonian flow of $$\sigma(\Delta_{g})$$ is the geodesic flow of $$g,$$ and therefore the previous comments apply.