Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems

Let (M, g) be a compact Riemannian manifold of dimension n and $$ so that $$ on $$ . We assume that $$ is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators $$ with $$ , $$ . We study the pointwise bounds for the joint eigenfunctions, $$ of the system $$ with $$ . In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in M. In two and three dimensions, these estimates agree with the Hardy exponent $$ and in higher dimensions we obtain a gain of $$ over the Hörmander bound. In our second main result (Theorem 3), under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points $$ in the "microlocally forbidden" region $$ These bounds are sharp locally near the projection of the invariant tori.