Restriction of Schrödinger eigenfunctions to submanifolds

Abstract

Burq-G\'erard-Tzvetkov and Hu established $L^p$ estimates for the restriction of Laplace-Beltrami eigenfunctions to submanifolds. We investigate the eigenfunctions of the Schr\"odinger operators with critically singular potentials, and estimate the $L^p$ norms and period integrals for their restriction to submanifolds. Recently, Blair-Sire-Sogge obtained global $L^p$ bounds for Schr\"odinger eigenfunctions by the resolvent method. Due to the Sobolev trace inequalities, the resolvent method cannot work for submanifolds of all dimensions. We get around this difficulty and establish spectral projection bounds by the wave kernel techniques and the bootstrap argument involving an induction on the dimensions of the submanifolds.