On the number of nodal domains of the 2D isotropic quantum harmonic oscillator -- an extension of results of A. Stern --

TL;DR

This paper extends previous results on the number of nodal domains of eigenfunctions to the 2D isotropic quantum harmonic oscillator, demonstrating the existence of eigenfunctions with two nodal domains for an infinite sequence of eigenvalues.

Contribution

It provides the first proof of the existence of eigenfunctions with two nodal domains for infinitely many eigenvalues in the 2D isotropic quantum harmonic oscillator.

Findings

Existence of an infinite sequence of eigenfunctions with two nodal domains.

Extension of classical results from spheres and squares to the quantum harmonic oscillator.

New methods for analyzing nodal domains in quantum systems.

Abstract

In the case of the sphere and the square, Antonie Stern (1925) claimed in her PhD thesis the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with two nodal domains. These two statements were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom--Przybytkowski (2006)). The aim of this paper is to obtain a similar result in the case of the isotropic quantum harmonic oscillator in the two dimensional case.