Quantum Flux and Quantum Ergodicity for Cross Sections

For sequences of quantum ergodic eigenfunctions, we define the quantum flux norm associated to a codimension $1$ submanifold $\mathrm{\Sigma }$ of a non-degenerate energy surface. We prove restrictions of eigenfunctions to $\mathrm{\Sigma }$, realized using the quantum flux norm, are quantum ergodic. We compare this result to known results from \cite{CTZ} in the case of Euclidean domains and hyperfurfaces. As a further application, we consider complexified analytic eigenfunctions and prove a second microlocal analogue of \cite{CTZ} in that context.