Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the velocity generated by this stream function. Suppose all the orbits generated by such a flow are periodic.
Let $L^0$ be the operator of differentiation along streamlines, i.e $L^0 g = - \mathbf u^0 \cdot \nabla g$ for all $g \in H^1$ where $L^0$ acts as an unbounded operator in $L^2_a$ the set of $L^2$ functions with zero space average.
If all the orbits generated by the steady state flow is periodic, prove that $0$ is an isolated eigenvalue in the spectrum of $L^0$.
i) Now it is clear that any function $\phi$ which is constant along streamlines of $\psi^0$ automatically satisfies $L^0 \phi = 0$ by which we get that $0$ is an eigenvalue. (Is this right?)
ii) How does one prove that $0$ is isolated, i.e with no accumulation points.
iii) Is it possible to characterize the essential spectrum of such an operator and perhaps also the point spectrum.
Thank you.