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

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 ${\int }_{\gamma }|{\varphi }_j^{\hslash }{|}^{2}ds=\mathcal{O}(|\log \hslash |)$ 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.

Correct some typos and added some more detail in section 2