Pointwise bounds for joint eigenfunctions of quantum completely integrable systems

TL;DR

This paper establishes improved pointwise bounds for joint eigenfunctions of quantum completely integrable systems on compact manifolds, including polynomial improvements in general and exponential decay outside invariant tori under real-analyticity.

Contribution

It provides new polynomial bounds for eigenfunctions at typical points and exponential decay estimates outside invariant tori in quantum integrable systems.

Findings

Polynomial improvements over Hörmander bounds at typical points.

Exponential decay of eigenfunctions outside invariant tori under real-analyticity.

Bounds are sharp near the projection of invariant tori.

Abstract

Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2\Delta_g+V(x)-E_1$ so that $dp_1\neq 0$ 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 $[P_i,P_j]=0$, $i,j=1,\dots ,n$. We study the pointwise bounds for the joint eigenfunctions, $u_h$ of the system $\{P_i\}_{i=1}^n$ with $P_1u_h=E_1u_h+o(1)$. 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.