Nodal sets of Schr\"odinger eigenfunctions in forbidden regions

Abstract

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.