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

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

2008-03-06

arxiv.org/abs/0803.0978v2

Mathematics

Analysis of PDEs

Correct some typos and added some more detail in section 2

Scientific paper

10.1007/s00220-009-0747-y

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.

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