Concentration of quantum integrable eigenfunctions on a convex surface of revolution

Abstract

Let $(S^2,g)$ be a convex surface of revolution and $H\subset S^2$ the unique rotationally invariant geodesic. Let ${\phi }_m^{\ell }$ 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, ${\Sigma }_{m=-\ell }^{\ell }||{\phi }_m^{\ell }|{|}_{L^2(H)}^2{\delta }_{\frac{m}{\ell }}(c)$ 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 ${\sum }_{m=-\ell }^{\ell }⟨B{\phi }_m^{\ell },{\phi }_m^{\ell }⟩{\delta }_{\frac{m}{\ell }}(c)$.

26 pages

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