Geometric structures in the nodal sets of eigenfunctions of the Dirac operator

Abstract

We show that, in round spheres of dimension $n\ge 3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{{\Sigma }_1,...,{\Sigma }_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $\psi =({\psi }_1,...,{\psi }_N)$ of the Dirac operator such that each submanifold ${\Sigma }_a$, modulo ambient diffeomorphism, is a structurally stable nodal set of the spinor component ${\psi }_a$. The result holds for any choice of trivialization of the spinor bundle. The emergence of these structures takes place at small scales and sufficiently high energies.

21 pages