Dislocations of arbitrary topology in Coulomb eigenfunctions

Abstract

For any finite link $L$ in ${\mathbb{R}}^3$ we prove the existence of a complex-valued eigenfunction of the Coulomb Hamiltonian such that its nodal set contains a union of connected components diffeomorphic to $L$. This problem goes back to Berry, who constructed such eigenfunctions in the case where $L$ is the trefoil knot or the Hopf link and asked the question about the general result.