L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case

TL;DR

This paper establishes logarithmic bounds on the restriction of eigenfunctions along curves in quantum integrable systems and identifies cases where maximal bounds are achieved.

Contribution

It provides the first logarithmic restriction bounds for eigenfunctions in quantum integrable systems and characterizes exceptional subsequences attaining maximal bounds.

Findings

Restriction bounds are of order |log ħ| for generic curves.

Maximal restriction bounds are achieved by certain eigenfunction subsequences.

Results apply specifically to two-dimensional quantum completely integrable systems.

Abstract

We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $ \int_{\gamma} |\phi_{j}^{\hbar}|^2 ds = {\mathcal O}(|\log \hbar|)$ for generic curves $\gamma$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions.