  This note concerns the nodal sets of eigenfunctions of semiclassical
Schr\"odinger operators acting on compact, smooth, Riemannian manifolds, with
no boundary. We prove that if H is a separating hypersurface that lies inside
the classically forbidden region, then H cannot persist as a component of the
zero set of infinitely many eigenfunctions. In addition, on real analytic
surfaces, we obtain sharp upper bounds for the number of intersections of the
zero sets of the Schr\"odinger eigenfunctions with a fixed curve that lies
inside the classically forbidden region.
