Pointwise bounds for joint eigenfunctions of quantum completely integrable systems

Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2{\mathrm{\Delta }}_g+V(x)-E_1$ so that $d{p}_1\mathrm{\ne }0$ on $p_1=0$. We assume that $P_1$ is quantum completely integrable in the sense that there exist functionally independent pseuodifferential operators $P_2,\dots P_n$ with $[P_i,P_j]=0$, $i,j=1,\dots ,n$\mathrm{.}Westudythepointwiseboundsforthejointeigenfunctions,$u_h$ of the system $\{P_i{\}}_{i=1}^n$ with $P_1{u}_h=E_1{u}_h+o(1)$. We first give polynomial improvements over the standard H"ormander bounds for typical points in $M$. In two and three dimensions, these estimates agree with the Hardy exponent $h^{-\frac{1-n}{4}}$ and in higher dimensions we obtain a gain of $h^{\frac{1}{2}}$ over the H"ormander bound. In our second main result, 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 $x\in M$ in the “microlocally forbidden” region $p_1^{-1}(E_1)\cap \dots \cap p_n^{-1}(E_n)\cap T^*_xM=\emptyset.$ These bounds are sharp locally near the projection of the invariant tori.