Nodal sets of Schr\"odinger eigenfunctions in forbidden regions

TL;DR

This paper investigates the behavior of nodal sets of semiclassical Schr"odinger eigenfunctions in forbidden regions, showing certain hypersurfaces cannot be zeros of infinitely many eigenfunctions and providing bounds on intersections with curves.

Contribution

It proves that separating hypersurfaces inside forbidden regions cannot be zeros of infinitely many eigenfunctions and establishes sharp bounds on zero set intersections on real analytic surfaces.

Findings

Hypersurfaces inside forbidden regions are not persistent zero sets for infinitely many eigenfunctions.

Sharp upper bounds are established for intersections of eigenfunction zeros with curves in forbidden regions.

Results apply to eigenfunctions on compact, smooth Riemannian manifolds without boundary.

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.