Trace estimates and improved pointwise bounds for joint eigenfunctions

TL;DR

This paper improves bounds on joint eigenfunctions in quantum integrable systems, providing sharper estimates at points with specific non-degeneracy conditions.

Contribution

It establishes a sharp bound for joint eigenfunctions at points satisfying a rank non-degeneracy condition, refining previous polynomial bounds.

Findings

Established a sharp bound of h^{(-n+k+1)/2} for certain points.

Improved upon previous polynomial bounds for typical points.

Focused on points with a rank k non-degeneracy condition.

Abstract

For $L^2$-normalized joint eigenfunctions in a quantum integrable system, [GT20] gave polynomial improvements over the standard H\"omander bounds for typical points. In this paper, we improve their result by establishing a sharp bound of $h^{\frac{-n+k+1}2}$ for the points satisfying a rank $k$ non-degeneracy condition.