Average estimate for Eigenfunctions along geodesics in the quantum completely integrable case

TL;DR

This paper establishes a new polynomial decay rate for integrals of eigenfunctions over geodesics in quantum integrable systems, improving upon previous bounds.

Contribution

It provides a rigorous proof of an asymptotic decay rate of O(h^{1/2}) for eigenfunction integrals over geodesics, advancing understanding in quantum integrable systems.

Findings

Asymptotic decay rate of O(h^{1/2}) for eigenfunction integrals

Improvement over previous O(1) bounds

Applicable to admissible geodesics in 2D quantum integrable systems

Abstract

This paper investigates the upper bound of the integral of $L^2$-normalized joint eigenfunctions over geodesics in a two-dimensional quantum completely integrable system. For admissible geodesics, we rigorously establish an asymptotic decay rate of $O(h^{1/2})$. This represents a polynomial improvement over the previously well known $O(1)$ bound.