Concentration of quantum integrable eigenfunctions on a convex surface of revolution

Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $\varphi^\ell_m$ be the orthonormal basis of joint eigenfunctions of $\Delta_g$ and $\partial_\theta$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $\Sigma_{m = -\ell}^\ell ||\varphi^\ell_m||^2_{L^2(H)} \delta_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$, and exhibit a $(1 - c^2)^{-\frac{1}{2}}$ type singularity at the endpoints. For a pseudo-differential operator $B$ we also compute the limits of the normalized measures $\sum_{m = -\ell}^\ell \langle B \varphi^\ell_m , \varphi^\ell_m \rangle \delta_{\frac{m}{\ell}}(c)$.