  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.
