A problem of Berry and knotted zeros in the eigenfunctions of the harmonic oscillator

TL;DR

This paper proves that for any finite link in three-dimensional space, there exists a high-energy eigenfunction of the harmonic oscillator whose nodal set contains components diffeomorphic to that link, solving a problem posed by Berry.

Contribution

It demonstrates the existence of eigenfunctions with knotted zeros corresponding to any finite link, advancing understanding of quantum eigenfunction topology.

Findings

Existence of eigenfunctions with knotted nodal sets for any finite link

Resolution of Berry's problem on knotted zeros in quantum bound states

Construction of eigenfunctions with prescribed topological features

Abstract

We prove that, given any finite link L in R^3, there is a high energy complex-valued eigenfunction of the harmonic oscillator such that its nodal set contains a union of connected components diffeomorphic to L. This solves a problem of Berry on the existence of knotted zeros in bound states of a quantum system.