Dislocations of arbitrary topology in Coulomb eigenfunctions

TL;DR

This paper proves that for any finite link in three-dimensional space, there exists a high-energy hydrogen atom eigenfunction whose nodal set contains components diffeomorphic to that link, extending previous special cases.

Contribution

It generalizes Berry's earlier work by showing the existence of eigenfunctions with nodal sets containing arbitrary finite links.

Findings

Existence of eigenfunctions with prescribed link-shaped nodal components

Extension of Berry's special case results to arbitrary links

Construction of high-energy eigenfunctions with complex topologies

Abstract

For any finite link $L$ in $\mathbb{R}^3$ we prove the existence of a high-energy complex-valued eigenfunction of the hydrogen atom 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.