Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems

Let (M, g) be a compact Riemannian manifold of dimension n and [Formula: see text] so that [Formula: see text] on [Formula: see text] . We assume that [Formula: see text] is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators [Formula: see text] with [Formula: see text] , [Formula: see text] . We study the pointwise bounds for the joint eigenfunctions, [Formula: see text] of the system [Formula: see text] with [Formula: see text] . 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 [Formula: see text] and in higher dimensions we obtain a gain of [Formula: see text] 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 [Formula: see text] in the “microlocally forbidden” region [Formula: see text] These bounds are sharp locally near the projection of the invariant tori.