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

الأصل · EN

We show that, in round spheres of dimension n≥3, for any given collection of codimension 2 smooth submanifolds S:={Σ₁,...,Σₙ} of arbitrarily complicated topology (N being the complex dimension of the spinor bundle), there is always an eigenfunction ψ=(ψ₁,...,ψₙ) of the Dirac operator such that each submanifold Σₐ, modulo ambient diffeomorphism, is a structurally stable nodal set of the spinor component ψₐ. 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.